Chapter 1. Crop model building and simulation in Microsoft Excel:
Introducing BuildIt
1.1
Spreadsheets for mathematical modeling
Mathematical modeling often incurs a steep learning curve to many
agriculturists. This is partly because one of the stages in the modeling process
involves computer programming, a skill many agriculturists lack. Consequently, many
agriculturists struggle to translate their formulated models into a set of instructions
that can be implemented by a computer system. For them, computer programming
often degenerates into a tedious and distracting undertaking. This problem has been
recognized even as early as three decades ago when Hillel (1977) commented that
agriculturists, being novice programmers, are easily entangled in the complexities of
translating their model into a computer source code. Rather than focusing on the
important—and intended—task of mathematical modeling and simulation, they
instead become distracted by the intricacies and mechanism of computer
programming.
What is required then is some special software that allows non-programmers
to build their own computer models. This will enable non-programmers, agriculturists
included, to accomplish their modeling tasks directly and quickly and without the
distraction of having to learn a new computer language (or to master a new simulation
application).
There are generally four categories of software platforms to aid in model
building and simulation. They are: 1) general purpose computer languages (e.g., C,
C++, FORTRAN, BASIC, Pascal and Java); 2) specialized simulation applications
(e.g., FST, PowerSim, Stella and ExtendSim); 3) equation solver-based applications
(e.g., Maple, Mathematica and Mathlab); and 4) spreadsheet-based applications (e.g.,
Microsoft Excel and OpenOffice.org Calc)1. Of these four groups of software
platforms, spreadsheet applications require the least level of proficiency in computer
programming.
It is no surprise then that spreadsheets have been advocated by several workers
(e.g., Seila, 2005; Brown, 1999; Nardi and Miller, 1990) as a suitable platform,
especially for non-programmers, to build and simulate their models. Spreadsheets are
immensely popular and widely used because they are easy to use and versatile.
Additionally, they provide the following features (adapted from Seila, 2005): 1) a
large number of numerical and non-numerical functions (i.e., dedicated formulas) to
do mathematical, statistical, database, date and time, financial, engineering and other
types of calculations; 2) database representations and access; 3) charting and
graphing; 4) display and document formatting capabilities such as layout, fonts and
colors to improve presentation; and 5) scripting or programming language such as
1
FST (Fortran Simulation Translator) is available upon request from its developers (see van
Kraalingen et al., 2003); Powersim is a registered trademark of Powersim Software AS; Stella is a
registered trademark of isee systems; ExtendSim is a registered trademark of Imagine That Inc.;
Maple is a registered trademark of Waterloo Maple Inc.; Mathematica is a registered trademark of
Wolfram Research Inc.; Mathlab is a registered trademark of The MathWorks Inc.; Microsoft
Excel are registered trademarks of Microsoft Corporation; OpenOffice.org Calc is a registered
trademark of OpenOffice.org
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VBA (Visual Basic for Applications) in Excel and OOoBasic (OpenOffice.org Basic)
in OpenOffice.org Calc. Spreadsheets were initially conceived as electronic
accounting books for financial analysis, but today, spreadsheets have progressed
beyond their original intent. They have become powerful tools to manipulate, analyze
and present data, and to build models for simulating various phenomena in science
and engineering (Khandan, 2001).
1.2
Advantages and disadvantages of spreadsheets
It is well known that spreadsheets are easy to use, versatile and powerful, but
it is not so much these characteristics that qualify spreadsheets as a suitable modeling
platform for non-programmers.
The first key benefit of using spreadsheets in modeling is that they do not
require users to be proficient in computer programming. This is achieved by the
spreadsheet system providing: 1) automatic control and maintenance of the program
flow; 2) a simple, straightforward modeling framework; and 3) high-level and taskspecific functions (Nardi and Miller, 1990).
A spreadsheet is a group of pages, or worksheets, where each worksheet is a
two-dimensional table consisting of rows and columns of cells. Each cell can contain
either a constant or a calculated value, derived from a function or formula. The open
tabular format of the spreadsheet provides users a simple, straightforward framework
in which to build their models. As such, spreadsheet users only need to understand
two basic concepts: to treat the cells as variables and the functions (or formulas) as
the relationship between these variables. Users specify the way variables depend on
each other via formulas or functions, and the spreadsheet system maintains these
variable dependencies. So if one part of the spreadsheet changes, it triggers an update
to the whole spreadsheet so that all the dependent variables are automatically
recalculated to reflect their new values—giving users immediate feedback. The order
in which the variables are calculated is worked out by the spreadsheet system based
on the variable dependencies. Users cannot contravene this calculation sequence by
giving the spreadsheet system a different calculation or action sequence to be
performed. Though this might seem restrictive, it is actually this feature, among
others, that makes spreadsheets appealing to non-programmers.
Spreadsheets not only relieve users from having to maintain the program flow
themselves, but also from having to design their own modeling framework. Flow
control and framework design are both difficult programming concepts for nonprogrammers to master (Nardi and Miller, 1990; Hoc, 1989; Lewis and Olson, 1987;
Soloway et al., 1983). Without automatic flow control, users will have to write their
own programming code to track variable dependencies and to update all effected
variables iteratively when required. And users need not design their own modeling
framework because the spreadsheet already provides users with one.
Spreadsheets further cater to non-programmers by providing high-level and
task-specific functions, meaning that these functions can be used without users having
to understand how they work or to require computer programming expertise. One
commonly used function in Excel, for example, is the SUM function that, as its names
implies, calculates the total of a set of given values. Spreadsheet users merely write
the SUM function and specify all the cells that contain the values to be added. Without
such a high-level function, users will require to write a loop to iterate through the
elements in an array, summing each element and accumulating the total. Other
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routinely used functions are such as AVERAGE (for calculating the mean of values),
MIN and MAX (for determining the smallest and largest value, respectively, from a
given set of values), VLOOKUP and HLOOKUP (for looking up a value from a
vertically and horizontally tabulated data, respectively), IF (for conditional
computations), and trigonometric functions like SIN and COS. These are only a few
of the functions listed; hundreds more functions are available in Excel (as well as in
other spreadsheets) to support users’ requirements.
The second key benefit of using spreadsheets is its table-oriented interface. It
provides a strong visual format to manipulate, organize and present data as well as a
problem-solving tool (Nardi and Miller, 1990). The spreadsheet’s open tabular format
allows direct access to the variables’ values. This is in contrast to traditional
programming efforts which require some special and intentionally constructed
mechanisms for the input and output of variables. Brown (1999) additionally noted
that because the values of all variables are always updated and displayed in the table,
this could help to detect the location of errors during model construction by the
presence of nonsensical or questionable values.
The spreadsheet’s tabular format additionally allows users to organize visually
their model into subparts or modules, where each module performs a specific task to
solve an overall, larger problem. This organization can be done by segmenting the
large model into smaller modules by, for instance, leaving an empty row between
each module or by placing each module in a separate worksheet. Formatting the
module sections such as by applying different fonts, shading, style layout or colors
further help in code organization. Consequently, the model’s code is distributed
visually over one or more grids in such a way that makes the model more
comprehensible and maintainable. Visual code organization is not merely for aesthetic
reasons. The very act of laying out data, organizing and formatting the code provide
an important visual feedback mechanism that helps to organize users’ thoughts and
shape their problem solving process (Nardi and Miller, 1990).
Spreadsheets, however, have several important limitations. Spreadsheet
models can be difficult to understand when studied by others. As mentioned
previously, the modeling framework provided by spreadsheets is a simple,
straightforward structure, essentially consisting of variables (cells) and their linkages
with each other (formulas). But as models increase in complexity, these variable
linkages can grow into an intricate network of relationships. Since the spreadsheet
system maintains the program flow and shields users from viewing this network of
relationships, it can be difficult for other users (even for the model developers
themselves) to grasp the computations as a whole. This causes difficulty not only in
understanding the model but also in debugging it (that is, to locate and correct the
errors in the model).
Spreadsheets offer to non-programmers a modeling platform that is almost
unrestrictive and freeform and with little validation checks. While this feature can be
convenient for those who wish to develop their models easily and quickly, it does,
however, make it easy for users to introduce errors unwittingly into their models.
Errors in large spreadsheet models, in particular, can be difficult to locate and, at
times, be undetected. It is well known in the business sector that the frequency of
errors in their spreadsheets are alarmingly high. Rajalingham et al. (2001) reported
that as much as 90% of real-world spreadsheets contained errors. Furthermore, the
European Spreadsheet Risks Interest Group (www.eusprig.org) lists nearly a hundred
3
examples of newsworthy incidences where spreadsheets errors caused companies
either financial losses, incorrect forecasting, or simply, embarrassment. Although
spreadsheets from the business sector are often cited as examples of high occurrence
of errors, we cannot disregard the possibility that spreadsheets from other disciplines
(such as agriculture) are too plagued by errors.
The simple modeling framework provided by the spreadsheet also makes
spreadsheet models difficult to reuse and extend. Reusability and extendibility are two
software engineering concepts related to ways to increase the usefulness, applicability
and lifespan of software. A modeling framework that adheres to these two concepts
means that, ideally, the framework is flexible and adaptable enough for algorithm
modification, substitution and addition. In some ways, reusability and extendibility
are analogous to the “plug-and-play” feature in modern computer systems. For
instance, adding, replacing or removing a computer peripheral (such as a printer,
scanner or mouse) should be a seamless operation that does not disrupt a computer
system or cause it to malfunction. In a similar way, a reusable and extendible model
means that one or more parts in the model can be modified easily or even substituted
with other parts from another model. It also means that additional features or
functions can be added to the model. All these operations can occur without having to
break or redesign the existing modeling framework.
There are several examples in agriculture of modeling frameworks developed
with concerns of achieving reusability and extendibility (e.g., Papajorgji and Pardalos,
2006; Papajorgji et al., 2004; Hillyer et al., 2003; Caldwell and Fernandez, 1998;
Acock and Reddy, 1997). Nevertheless, until today, none of them has yet to be
adopted widely by agriculturists. One could speculate that these framework designs
are still not sufficiently adaptable and flexible enough to cope with the unpredictable
and transient requirements of agricultural models. Or it may simply be that
agriculturists do not have the interest or time to learn about reusability and
extendibility, both concepts of software engineering, not agriculture. Consequently,
for many agriculturists, the problem that spreadsheet models are difficult to reuse and
extend may actually be an unimportant concern in their work.
Spreadsheets have other limitations that are particularly relevant for models
with complex data structures and algorithms (Seila, 2005). A spreadsheet is a twodimensional table of cells, and while this format is adequate in most cases, it is a
limitation when more elaborate data structures are required such as lists and trees.
Although spreadsheets can be coerced to handle such complex data structures, the
way they are achieved in spreadsheets can be inefficient and convoluted. Complex
algorithms can be particularly difficult to implement in a spreadsheet. Spreadsheets
lack explicit loop and conditional controls commonly found in general purpose
languages. Spreadsheets actually have a conditional construct IF-THEN-ELSE but its
effect is only local to the cell that contains the construct. The construct cannot be used
to change the values in other cells or to transfer control to another cell or to another
part of the spreadsheet. As stated earlier, spreadsheets relieve users from having to
maintain the control flow themselves. While this is a convenient feature for nonprogrammers, it becomes a severe limitation when an algorithm requires users to
specify explicitly the sequence of actions to be performed. Spreadsheets like Excel
and OpenOffice.org Calc have their own scripting language that can be used to
circumvent this problem, but they require proficiency in computer programming. But
non-programmers may find this solution unappealing because their use of
spreadsheets is precisely to avoid computer programming in the first place.
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Lastly, spreadsheet models often run slower than models developed in other
software platforms. This is partly because models developed in other platforms, such
as in C, C++ or FORTRAN, are compiled (translated) into fast machine code that is in
a format directly executable by the computer. In contrast, no such compilation occurs
for spreadsheet models. The formulas in spreadsheets are instead interpreted first
before execution. This slows down program execution. Dalton (2005) roughly
estimated that models written in Excel could be slower by as much as five to twenty
times than if they were written in C and C++ computer languages. He further reported
that Excel’s text-manipulation routines are very much slower than the same routines
in C and C++ by as much as over 300 times. Slow execution speeds may not be
noticeable for simple spreadsheet models, but for larger, more complex models (due
to their more numerous and intricate calculations), their execution speeds can be
noticeably sluggish and become a significant constraint.
For non-programmers, spreadsheets offer an infrastructure that appeals to
them because spreadsheets allow them to build fully functioning models quickly and
directly, and without the pre-requisite knowledge in any advance programming
concepts and techniques. But for those with programming expertise, the limitations of
spreadsheets can be significant reasons why spreadsheets are not used as their
modeling platform. In some cases, spreadsheets are merely used as a quick test bed to
test their modeling ideas or to develop model prototypes. Once these ideas or model
prototypes have been assessed sufficiently, model development then shifts to other
software platforms for completion.
1.3
How to use BuildIt
BuildIt is an addin that works within Excel to support model building and
simulation in Excel. It is important to realize that BuildIt is neither a crop model nor a
“toolbox” of useful equations. The primary purpose of BuildIt is to overcome some
limitations in Excel by providing a framework in particular for non-programmers to
build their models, especially those that are large and complex and those that have
iterative calculations.
Crop models, for instance, are often large and complex because they comprise
several interrelated components such as meteorology, water and energy balance, and
plant photosynthesis and respiration. Crop models also often involve iterative
calculations, where the same set of calculations are repeated over and over to simulate
the dynamics of plant growth and its environmental conditions.
1.3.1 Installing BuildIt
BuildIt can be obtained free from: www.agri.upm.edu.my/~chris/buildit.
Download the archive file buildit.zip. The archive contains two file versions for
BuildIt: one that works in Excel 2003 (buildit.xla) and another for Excel 2007
(buildit.xlam).
Excel 2003
Extract the buildit.xla file and save it a folder in your computer. You can save
the buildit.xla in any folder, but by default, Excel addins are saved in several
locations, one of which is in the folder
Documents and Settings/user name/Application Data/Microsoft/AddIns
where user name is your user name for your computer.
5
To use BuildIt, you need to load it in Excel. Start up Excel, and from the main
menu, choose Tools|Add-ins. You will see a dialog box, listing all available addins
(Fig. 1.1). If you do not see BuildIt’s name in the list, click the Browse button to
search for the file buildit.xla. Once located, click OK and the BuildIt’s name should
be listed in the dialog box. Check (tick) the box next to BuildIt’s name and dismiss
the dialog box by clicking the OK button. Once loaded, BuildIt will install its own
menu on the main menu bar (Fig. 1.2). BuildIt is now loaded in Excel and ready for
use each time you start up Excel.
Fig. 1.1. Load BuildIt by ticking its checkbox and clicking the OK button. Note that
the list of available addins will differ from one computer system to another.
Fig. 1.2. BuildIt menu in Excel 2003
Excel 2007
Extract the buildit.xlam file and save it a folder in your computer. You can
save the buildit.xlam in any folder, but by default, Excel addins are saved in several
locations, one of which is in the folder
Documents and Settings/user name/Application Data/Microsoft/AddIns
where user name is your user name for your computer.
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To load BuildIt in Excel2007, first click the Microsoft Office Button
,
and then click Excel Options. Click the Add-Ins category. In the Manage box, click
Excel Add-ins, and then click Go (Fig. 1.3). To load BuildIt, select the check box next
to BuildIt’s name in the Add-Ins available box (Fig. 1.1). If you do not see BuildIt’s
name in the list, click the Browse button to search for the file buildit.xlam. Once
located, click OK and the BuildIt’s name should be listed in the dialog box. Dismiss
the dialog box by clicking OK button. Once loaded, BuildIt will install its own ribbon
(Fig. 1.4). BuildIt is now loaded in Excel and ready for use each time you start up
Excel.
BuildIt commands
Fig. 1.2 and 1.4 show that there are four commands in the BuildIt menu:
1. Reset model: Performs the operations in the Prerun section. It also sets _step
to 0.
2. Do operations only: Performs the operations in the Operation section.
3. Start simulation: Runs the model.
4. About BuildIt: Displays program information.
Unloading and uninstalling BuildIt
BuildIt can be unloaded from Excel by merely removing the check mark from
the list of available Add-Ins (Fig. 1.1). If you wish to uninstall (remove completely)
BuildIt, first unload it from Excel as previously described, then simply delete the
BuildIt file from your computer.
Fig. 1.3. Available addins in Excel 2007
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Fig. 1.4. BuildIt ribbon in Excel 2007
1.3.2 BuildIt’s framework and functions
Fig. 1.5 shows BuildIt’s framework and program flow in Excel. When
simulation begins, BuildIt checks the Option section. A section is a range of cells in
the spreadsheet (workbook) that lists one or more operations to be executed by
BuildIt. We create a section by defining a chosen cell with a specific name, and this
cell becomes the start of that section. The Option section, for example, is created
when we define the name _option to one of the worksheet cells in our workbook. At
this point, you may wonder what exactly is an operation and how they are specified
for BuildIt. These will be explained later, but for the moment, just know that an
operation is some specific task to be performed by BuildIt in our spreadsheet. In a
given section, we list the operations in the order (from top to bottom) we want them to
be executed by BuildIt. So if we want operation A to be executed before B and C, we
then list operation A, B then C in that order.
So the Option section is one portion in your worksheet that lists all the
operations you want to be done at the beginning of a simulation run. We see from Fig.
1.5 that the operations in the Option section are only done once for each simulation
run. The Option section is optional; that is, if we do not define the name _option then
BuildIt proceeds into the outer loop.
As indicated by the dashed boxes in Fig. 1.5, there are two loops in BuildIt’s
program flow: the outer and inner loops. The outer loop controls the scenario runs.
Every simulation run has at least one scenario. As an example, let us take a trivial
equation for a model:
y  mx  c
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where m is the slope of the line and c is the intercept on the y-axis. We fix c as 0 and
m as 1. We wish to simulate (or calculate, in this case) the values for y for x = 0, 1, 2,
3, …, 10. This is our only scenario so BuildIt runs our simple model, calculates and
outputs the results, and simulation ends. The outer loop runs once since we have only
one scenario.
Fig. 1.5. Framework and program flow for BuildIt
Let us say that we are also interested to determine how y varies with x with m
= 2 and c = 1. This is in addition to the earlier values of m = 1 and c = 0. We can of
course run the model twice: once for m = 1 and c = 0 and then for m = 2 and c = 1.
9
BuildIt, however, provides the support where we can run these two scenarios (or
more) in a single simulation run.
We define one of the worksheet cells with the name _scenario to denote the
start of the Scenario section and to instruct BuildIt that there is more than one
scenario in the simulation run. In this example, BuildIt runs the model for m = 1 and c
= 0 first, then for m = 2 and c = 1, giving two output listings, one for each scenario.
Again, how we specify the instructions for running two or more scenarios will be
discussed later.
Unlike the optional names _option and _scenario, it is compulsory that we
define one of the worksheet cells with the name _step. This is because BuildIt always
sets the _step variable to 0 in the beginning of the outer loop (Fig. 1.5). _step can be
regarded like an iteration counter for the inner loop, but before BuildIt proceeds into
the inner loop, it checks for the Prerun section. This section is created when we
define one of the worksheet cells with the name _prerun. Though this section is
optional, it usually required to initialize any model parameters. In other words, one of
the uses of the Prerun section is to instruct BuildIt which are the model parameters to
be initialized and what values to give them.
The inner loop forms the hub of the calculations. Together with _step, we
must also define the names _stepsize and _criteria. The name _criteria states
the logical condition to which to end the cyclical calculations in the inner loop. As
long as _criteria is TRUE, the iteration in the inner loop continues until _criteria
becomes FALSE, after which BuildIt exits into the outer loop.
Two key activities occur in the inner loop. The first is model output, and the
second is the execution of BuildIt operations listed in the Operation section. This
section is created by defining the name _operation to one the worksheet cells.
Though the Operation section is optional, this section is often crucial to allow certain
calculations in the model to occur. As indicated in Fig. 1.5, there are two points for
model output: one before the operations in the Operation section are executed, and
another point after their executions. There are two points for model output because the
values of some variables will be changed after the operations in the Operation section.
Therefore, their values before and after these operations will be different.
The output of model simulation results require that we define the names _read
and _write. A cell defined with the name _read is where we specify which variables
are to be saved in the model output and whether they should be saved before or after
the operations from the Operation section are executed. Complementing _read, is
_write which specifies the starting cell where the model output should be stored.
Every iteration cycle in the inner loop ends by incrementing _step by
_stepsize, where _stepsize is the interval between two iteration steps. The logical
condition of _criteria is checked to determine if the next iteration cycle should
continue or stop.
When BuildIt exits into the outer loop, all operations, if any, in the Postrun
section (starts with the cell defined with _postrun) are executed. In most cases,
however, there is seldom a need for the Postrun section.
BuildIt then checks if there is any more modeling scenarios. If there is, BuildIt
changes some values or formulas as instructed in the Scenario section (recall that it is
the cell defined with _scenario), _step is reset to 0 again, and as before, all the
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model parameters will be initialized as instructed in the Prerun section. This whole
process, as discussed above, repeats until all scenarios have been exhausted.
Because calculations can be lengthy, BuildIt provides a feature where
calculations can be aborted. Pressing the Escape key or Control-C keys will abort the
simulation run. At times, these keys have to be pressed repeatedly because Excel may
ignore key presses during intensive processes or calculations.
Table 1.1 summarizes the names recognized by BuildIt and their brief
description. All names recognized by BuildIt start with an underscore.
Table 1.1. List of cell names recognized by BuildIt. Mandatory names are those that
must be defined.
Name
Description
_step
Iteration step (mandatory)
_stepsize
Interval between each iteration step (mandatory)
_criteria
Condition for ending the iteration in the inner
loop (mandatory)
_read
Location for specifying model variables that are
to be saved
_write
Location for saving the model simulation results
_operation
Indicates the start of the Operation section
_prerun
Indicates the start of the Prerun section
_postrun
Indicates the start of the Postrun section
_scenario
Indicates the start of the Scenario section
_option
Indicates the start of the Model Option section
The following describes BuildIt’s various operations. They are better
understood when we later see how they can be used in several examples. Nonetheless,
they are listed below in one place as reference.
ITG operation
ITG (x, func, output, lower, upper, single_integral)
Integrates the function func of x over the interval [lower, upper], and the
result of the integration is to be stored in the output cell. ITG for single integrations
can be represented as:
upper

output 
func( x)dx
lower
ITG’s
last argument, single_integral is either TRUE for single integrations or
FALSE for multiple integrations (e.g., double or triple integrations). The argument
single_integral can be omitted and will be assumed TRUE by default.
INI operation
INI (destination, source)
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Copies the contents from the source cells to the cells being referred to in the
destination cells. This operation is used primarily for setting initial values to
variables.
REP operation
REP (destination, source, num_of_replacements)
Copies the contents from source cells to the cells being referred to in the
cells as many as num_of_replacements times.
destination
ACC operation
ACC (x, accumulated_x, operator)
Sums or multiplies a series of numbers, depending on whether operator is
either + or * for addition or multiplication, respectively. If the operator is addition
(+), ACC sums the numbers:
accumulated_x 
 xi
i
But if the operator is multiplication (*), ACC multiplies the numbers:
accumulated_x 
 xi
i
RUN operation
RUN macro_name
Executes the macro macro_name. Macros can be a simple sequence of
recorded key strokes or actions to be done, or they can be quite sophisticated,
requiring programming proficiency in Visual Basic for Applications (VBA).
Developing macros, however, is beyond the subject of this book.
BuildIt, however, supplies three utility macros:
1. ClearOutput
Deletes the previous model output
2. DisableScreenUpdate
Switches off the automatic screen update
3. EnableScreenUpdate
Switches on the automatic screen update
The first macro, ClearOutput, is useful to clear the previous model output. Model
simulations are often run many times, with each run producing its own output result.
Consequently, in the Output section, you may have an overlap of the latest and old
model output. This macro is often used in the Option section so that before a
simulation begins, the old output is cleared entirely. The following is one way to use
this macro:
Row
1
2
Col. J
OPTION
run
Col. K
clearoutput
The Option section begins at cell J2 (defined with the name _option). Cell J2
specifies to run a macro whose name is specified in cell K2. Notice that the macro
name is case-insensitive.
The other two macros, DisableScreenUpdate and EnableScreenUpdate,
concern the computer screen refresh. Turning off the screen update speeds up the
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simulation run because Excel will not refresh the cells values. In other words, the
computer screen appears “frozen”. Though appearing to be stalled, the simulation is
still running and cells are still being updated but the computer screen does not refresh
to view their new updated values. Turning on the screen update will then “unfreeze”
the screen and all cells that are visible on the screen will have their correct and current
values. These two macros are useful in particular for models which have intensive and
lengthy calculations.
DisableScreenUpdate is often used in the Option section so that the screen
update is disabled before the lengthy calculations begin. Below is one way it can be
specified:
Row
1
2
Col. J
OPTION
run
Col. K
disablescreenupdate
This macro can also be used before a lengthy operation. The screen update is disabled
before a lengthy operation and reenabled after the operation completes. Consider the
following entries in the Operation section (with cell J2 defined with the name
_operation):
Row
1
2
3
4
Col. I
OPERATION
off screen update
lengthy REP op.
on screen update
Col. J
Col. K
Col. L
Col. M
run
rep
run
disablescreenupdate
=D3:H3
enablescreenupdate
=D4:H4
1000
Cells J3 to M3 specify that the contents in the range of cells D4 to H4 are to be copied
to cells D3 to H3 as many as 1000 times. This can be a lengthy operation because
each time cells D4:H4 are copied to cells D3:H3, it triggers an update involving
possibly lengthy and intensive calculations throughout the spreadsheet. Do this 1000
times and simulation speed can be very slow. To speed up the simulation run, the
screen update is disabled (cells J2 and K2) before the REP operation in cells J3 to M3.
When the REP operation completes, the screen update is reenabled in cells J4 and K4.
You do not have to worry about forgetting to reenable the screen update after
disabling it because BuildIt always ensures that the screen update is enabled before a
simulation ends or even when a simulation run is aborted (such as by pressing the Esc
or Control-C keys).
Lastly, BuildIt supplies one custom function and it is for linear interpolation.
Interpolate function
This function is called interpolate and it has the following specification:
interpolate (array_x, array_y, find_x)
Based on an array of given (x, y) values, the interpolate function returns y
corresponding to the given find_x. If no exact match is found, this function performs
a linear interpolation to obtain y. It is important to remember that array_x must be
sorted ascendingly (from smallest to largest value) before this function can be used
reliably.
Consider the following data set in Fig. 1.6 where the interpolate function is
used to lookup several values.
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Fig. 1.6. Data table to demonstrate the use of BuildIt’s interpolate function
Based on the tabulated data in Fig. 1.6:
interpolate(A2:A10, B2:B10, 2)
returns 20
interpolate(A2:A10, B2:B10, 3)
returns 35
interpolate(A2:A10, B2:B10, 20)
returns #VALUE!
The first case returns the exact y value of 20 because an exact match was found. The
second case fails to find an exact match, so this function linear interpolates between
(2, 20) and (4, 50) to give 35. The last case returns an error value because find_x lies
beyond the range of array_x.
Let us now consider several examples to better understand how BuildIt can be
used to build models.
1.3.3 Example 1: Leaf photosynthesis
As given by Goudriaan and van Laar (1994), leaf photosynthesis can be
described by the following equation:
AL 
Am kI 0 exp  kL 
Am   kI 0 exp  kL 
(1.1)
where AL is the leaf photosynthesis (that is, the assimilation rate of CO 2 by single
leaves; g CO2 m-2 leaf area s-1); Am is the maximum leaf photosynthesis rate (g CO2
m-2 leaf area s-1);  is the solar radiation conversion factor (g CO2 J-1); k is the
canopy extinction coefficient for solar radiation (unitless); I0 is the solar irradiance
above canopy (W m-2 ground area or J m-2 ground area s-1); and L is the cumulative
leaf area index from the canopy top to the canopy depth being considered (m2 leaf
area m-2 ground area).
In this example, we wish to determine how AL varies with L, with L increasing
from 0 to 10 m2 m-2. For this task, let us take Am as 1500 g CO2 m-2 s-1; k as 0.5; I0 as
200 W m-2; and  as 12 g CO2 J-1. All these equation variables are treated as
constants except for L which we will vary from 0 to 10 with an interval of, say, 0.5.
14
Begin by creating an Excel workbook and save it as Ex1. Rename one of the
worksheets as LeafPhotosyn. This is the worksheet where we will build Eq. (1.1).
Enter the following in the LeafPhotosyn worksheet:
Row
7
8
9
10
11
Col A
PARAMETERS
Am
k
Io
quantum
Col. B
Cell name
1500
0.5
200
12
B8: Am
B9: k
B10: Io
B11: e
The values of the variables are in cells B8 to B11. Cells A7 to A11 are merely labels
for documentation purposes and they are optional. Nevertheless, it is good
documentation practice to label or provide a brief description to all variables and
calculation steps. As required, we see that Am is 1500 g CO2 m-2 s-1 (cell B8); k is 0.5
(cell B9); I0 is 200 W m-2 (cell B10); and  is 12 g CO2 J-1 (cell B11). We also define
the following names to these cells: Am to cell B8, k to B9, Io to B10, and e to B11.
These names are convenient because they can be used in formulas rather than their
cell references to which they refer. For instance, instead of writing the formula like
this:
=E8*E11*E9*E10*EXP(-E9*3)
we can instead use their cell names to write:
=Am*e*k*Io*EXP(-k*3)
which makes reading and understanding the formula easier.
There are two ways to define a name to a cell. As an example, let us define the
name Am to cell B8. Place the active cell indicator at cell B8 in the Leaf Photosyn
worksheet (Fig. 1.7), then

Type Am in the Name box, and press Enter when done. Cell B8 will have the
name Am.

Alternatively, in Excel 2003, choose from the main menu Insert|Name|Define
and in the dialog box, type Am. In Excel 2007, from the Formulas ribbon, choose
Define name. In the dialog box, type Am. The name scope should be Workbook,
meaning that Am is visible (that is, can be referenced by other cells) in all
worksheets in the workbook.
Perform these same steps to define the names k, Io and e to cells B9, B10 and B11,
respectively.
Now we have to enter the following in the LeafPhotosyn worksheet:
Row
1
2
3
4
5
Col. A
CONTROL
max steps
step size
step
criteria
Col. B
Cell name
10
0.5
0
=_step<=B2
B3: _stepsize
B4: _step
B5: _criteria
It is compulsory that we define one cell with the name _step and another cell
_stepsize. In this example, we define cell B3 with the name _stepsize and cell B4
_step, and their values are 0.5 and 0, respectively. While the value of _stepsize is
15
important, the value we give to _step is not (we can even leave it blank). This is
because BuildIt maintains the value of _step. Before proceeding into the inner loop,
BuildIt resets _step to 0, and at the end of each iteration cycle in the inner loop,
_step will be incremented automatically by as much as _stepsize; that is
_stepi+1 = _stepi + _stepsize
where _stepi and _stepi+1 denote the value of _step at iteration no. i and the
subsequent step no. i+1, respectively (Fig. 1.5). We previously set _stepsize to 0.5
so when BuildIt enters into the inner loop, _step will have, as stated earlier, the value
0, then incremented to 0.5 in the next iteration cycle, then in successive iteration
cycles, incremented to 1.0, 1.5, 2.0 and so on.
Fig. 1.7. One way to define the name Am to cell B8 is to move the active cell indicator
to cell B8, type Am in the Name box, and press the Enter key
This progressive increase of _step will continue indefinitely unless we
specify a criteria to BuildIt so that it knows when to stop the iteration cycle in the
inner loop. In this example, cell B5 holds the criterion to end the simulation run, and
this cell is defined with the name _criteria. Like the case for _step and _stepsize,
it is also compulsory that we define one cell with the name _criteria. The formula
in cell B5 tells BuildIt that as long as _step<=B2 is TRUE, the iterative calculations
will continue until _step<=B2 becomes FALSE. Cell B2 is the upper limit (we gave it
the value 10) so when _step exceeds 10, the logical condition _step<=B2 becomes
FALSE which will end the inner loop calculations and BuildIt exits back into the
outer loop.
16
The purpose of having to setup the cells for _step, _stepsize and _criteria
will become evident in the following cell formulas to calculate the leaf
photosynthesis:
Row
1
2
3
4
5
Col. D
MODEL
L
n1
n2
AL
Col. E
Cell name
=_step
=Am*e*k*Io*EXP(-k*L)
=Am+e*k*Io*EXP(-k*L)
=E3/E4
E2: L
Cells E3 and E4 are the intermediary variables for the numerator and denominator in
Eq. (1.1), respectively. Cell E5 is the leaf photosynthetic rate at the given L (cell E2).
The formula in cell E2 (L) is set equal to _step. When BuildIt starts the
simulation run, _step is set to 0 and in turn, L is also 0. Using this L value, the leaf
photosynthetic rate (cell E5) is next calculated. In the next iteration cycle, _step is
incremented to 0.5 (L, likewise, is 0.5), and cell E5 produces the leaf photosynthetic
rate. This process continues for L = 1.0, 1.5, and so on until L exceeds 10 (the upper
limit which we set in cell B2). In other words, what we have done here is to determine
the leaf photosynthetic rate for L = 0 to 10 with a 0.5 interval. We can of course
choose a smaller or wider interval. For example, for a smaller and wider interval, we
can set _stepsize (cell B3) to 0.1 and 1.0, respectively, where the former interval
produces 101 points (L = 0, 0.1, 0.2,…, 10) and the latter interval only 11 points (L =
0, 1, 2,…, 10).
What if we wish to determine the leaf photosynthetic rate from L = 1 to 10,
instead of L = 0 to 10? In this case, we modify the formula in cell E2 from
=_step
to
=_step+1
so that L now starts at 1.0 and will be incremented in each subsequent iteration to 1.5,
2.0, 2.5, and so on until L = 10.
Calculations for L = 0 to 10 with a 0.5 interval produces 21 points, and we
need to save the model results for each point. For saving the model output, we need to
define one cell with the name _read and another cell with _write. The name _read
tells BuildIt which variables we wish to save, and the name _write tells BuildIt
where we wish to save these variables. In other words, _read can be regarded to mean
“read what?” and _write as “write where?”. The following entries illustrate the use of
both names:
Row
1
2
3
4
5
6
Col. G
to output
L
=L
Col. H
Cell name
AL
=E5
G3: _read
model output
G6: _write
Cells G1, G2, H2 and G5 are merely headers to document our entries and they are
optional. As before, it is recommended that we always label or provide a short
17
description for each entry in the spreadsheet for documentation purposes. The
important cells are G3 (defined with the name _read) and G6 (_write).
The cell defined with _read and all cells to its right will be checked by BuildIt
to determine which variables are to be saved in each iteration cycle. In our example,
the values referred to in cell G3 and in the cell to its right, cell H3, will be saved by
BuildIt. Cell E3 contains the value for L, and cell H3 the value of the corresponding
leaf photosynthetic rate. This means that BuildIt saves the values of L and AL in each
iteration cycle.
What if we also wish to save the numerator (n1) and denominator (n2) of Eq.
(1.1) for, say, checking purposes? For this purpose, we can enter the following:
Row
1
2
3
Col. G
to output
L
=L
Col. H
Col. I
Col. J
Cell name
AL
=E5
n1
=E3
n2
=E4
G3: _read
This means that BuildIt now saves the results referred to in cells G3 to J3. Recall that
the cell defined with _read serves as a starting point for BuildIt to determine which
variables are to be saved. All cells to the right of this starting cell will be checked by
BuildIt. However, these cells must not be blanks. BuildIt stops checking the moment
it encounters a blank cell. For example, the following entries cause BuildIt to save
only the values of L, AL and one of the intermediary variables, n1:
Row
1
2
3
Col. G
to output
L
=L
Col. H
Col. I
AL
=E5
n1
=E3
Col. J
Col. K
Cell name
n2
=E4
G3: _read
Cell J3 is blank so it causes BuildIt to stop checking and ignore cell K3. If there are
more entries to the right of cell K3, they too will be ignored (and not saved) by
BuildIt. In other words, variables to be saved must be specified in a continuous range
of cells, starting from the right of—and including—the cell defined with _read.
BuildIt actually checks two rows of cells starting from the cell defined with
_read. The first row is checked to determine which variables are to be saved, whereas
the second row specifies if the values of these variables are to be saved before or after
the operations in the Operation section are executed (Fig. 1.5). If they are to be saved
before these operations, the value FALSE must be given, else their values must be
TRUE. In this example, we earlier typed the following:
Row
1
2
3
4
5
6
Col. G
to output
L
=L
Col. H
Cell name
AL
=E5
G3: _read
model output
G6: _write
The first row checked by BuildIt is G3 and H3. This row, as stated before, tells
BuildIt which variables are to be saved in every iteration cycle in the inner loop (in
this case, L and AL). The second row checked by BuildIt is the one below; that is, G4
and H4. Both these cells are blanks (i.e., they are given no values). If a cell is left
blank, BuildIt assumes the default TRUE value, so both G4 and H4 are treated to have
18
TRUE values. This means that the values of L and AL are saved only after the
operations in the Operation section are executed. But if both G4 and H4 are FALSE,
their values then saved before these operations are executed.
In our example, however, it actually does not matter whether G4 or H4 is
TRUE or FALSE: model output remains the same. But for some variables, we may
need to specify whether their values should be saved before or after the Operation
section. One such a situation will be discussed later (section 1.3.6).
After specifying which variables are to be saved, we need to tell BuildIt where
these variables are to be saved. We define cell G6 the name _write, so cell G6 and all
cells to its right and all cells below it will be used by BuildIt to store the model
output. In our example, L and AL will be saved at location starting at cell G6. In the
first iteration, cells G6 and H6 will contain the values of L and AL, respectively. In the
next iteration, cells G7 and H7 will now store L and AL, respectively, and so on. In
other words, model output is a list that grows progressively downward where the
output of each iteration cycle occupies a separate row (Fig. 1.8).
We have completed implementing Eq. (1.1) in Excel and we are ready to run
the simulation. From the main menu, choose Start Simulation from the BuildIt
menu. The model output is as shown in Fig. 1.8. From the model output list (cells
G6:H26), we can draw a line scatter graph which shows an exponential decline in AL
with increasing L.
Fig. 1.8. Model for leaf photosynthesis: its setup and output
In Fig. 1.8, note that at the end of the simulation run, _step (as well as L) is
equal to 10.5 which exceeds the upper limit of 10 (cell B2), so the condition in cell B5
is FALSE, ending the simulation run. Model output begins in cells G6 and H6 and
19
moves progressively downward in every iteration cycle, ending at cells G26 and H26.
From the output list, we see that L (cell G6 downwards) begins at 0 and ends at 10
with a 0.5 interval, and its corresponding calculated AL is from H6 to H26.
Recall that we can press the ESC or Control-C keys to abort the simulation
run. This is particularly useful for lengthy runs. Note that you may have to press these
keys repeatedly to stop the simulation run. This is because Excel may ignore key
presses while it is performing intensive and lengthy calculations.
Eq. (1.1) is a simple equation and admittedly, it is actually unnecessary to use
BuildIt to implement Eq. (1.1) in Excel. The method as shown by Brown (1999) can
be used to implement such simple equations in Excel. The use of BuildIt becomes
essential, however, when we attempt to implement large, complex models in Excel, as
we will later see in this book. For now, we will continue examining more examples to
show how we can use BuildIt in model building and simulation.
1.3.4 Example 2: Canopy photosynthesis
Eq. (1.1) determines the leaf photosynthetic rate, but to determine the
photosynthetic rate of the whole canopy, we need to integrate Eq. (1.1) over the whole
canopy; that is
LAI
AT 

0
LAI


0
AL dL
Am  kI 0 exp  kL 
Am   kI 0 exp  kL 
(1.2)
dL
where AT is the canopy photosynthesis (g CO2 m-2 leaf area s-1); and LAI is the total
leaf area index of the whole canopy (m2 leaf area m-2 ground area). BuildIt solves
integration numerically using the 5-point Gaussian integration method.
Create a new workbook and save it as Ex2. Rename one of the worksheets as
CanopyPhotosyn. This is the worksheet where we will implement Eq. (1.2) and run it
to determine the canopy photosynthesis rate, AT, for LAI = 0 to 10 with a 0.5 interval.
As we have done in the previous example, enter the following for the Control
section in the CanopyPhotosyn worksheet:
Row
1
2
3
4
5
Col. A
CONTROL
max steps
step size
step
criteria
Col. B
Cell name
10
0.5
0
=_step<=B2
B3: _stepsize
B4: _step
B5: _criteria
The following are the entries for the equation variables (same as before):
Row
7
8
9
10
11
Col A
PARAMETERS
Am
k
Io
quantum
Col. B
Cell name
1500
0.5
200
12
B8: Am
B9: k
B10: Io
B11: e
20
And the section which implements Eq. (1.2) is as follows:
Row
1
2
3
4
5
6
7
Col. D
MODEL
LAI
L
n1
n2
AL
AT
Col. E
Cell name
=_step
E2: LAI
E3: L
=Am*e*k*Io*EXP(-k*L)
=Am+e*k*Io*EXP(-k*L)
=E4/E5
It may seem confusing that there seem to be two leaf area indexes: one in cell
E2 (LAI) and another in cell E3 (L). LAI is the total leaf area of the whole canopy per
unit ground area, whereas L is the cumulative leaf area (per unit ground area) from the
canopy top to a given canopy depth. For instance, at half the total canopy depth, L is
the sum of the leaf area from the canopy top to half the total canopy depth. So as we
move progressively down the canopy depth, L increases from 0 to LAI. In short, 0  L
 LAI.
Cells E4 and E5 are the numerator and denominator of Eq. (1.2), respectively,
and their ratio gives the leaf photosynthetic rate in cell E6. Note that both these cells
(E4 and E5) refer to cell E3 (L), but cell E3 is blank. It is left blank because BuildIt
will supply the necessary values for L during the numerical integration of Eq. (1.2).
As stated earlier, BuildIt uses a 5-point Gaussian integration method. This
means that five different values of L will be selected within the interval [0, LAI] by
BuildIt. And for each of the selected L value (cell E3), the corresponding AL (cell E6)
will be calculated. This gives a total of five different AL values, and they are summed
in such a way according to the Gaussian integration method to obtain the integration
result, which is stored in cell E7.
To perform an integration we use BuildIt’s integration operation, ITG. As
shown in Fig. 1.9, enter the following in cells J2 to P2:
Row
2
Col. J
Canopy photosynthesis
Col. K
itg
Col. L
=L
Col. M
=E6
Col. N
=E7
Col. O
=0
Col. P
=E2
Also define the name _operation for cell K2. This name indicates to BuildIt the start
of the Operation section where the list of operations are located. In our case, there is
only one operation in the list and this list begins in cell K2 onwards. Cell J2 is merely
for documentation purposes to describe the intention of the integration specification in
cells K2 to P2.
Fig. 1.9. BuildIt’s integration operation, ITG, and its specification. Note that the last
argument, single_integral, is left blank in cell Q2 so it is assumed TRUE by
default.
The ITG operation has the following specification:
21
ITG (x, func, output, lower, upper, single_integral)
which specifies that the function func of variable x is to be integrated over the
interval [lower, upper], and the result of the integration is to be stored in the output
cell. The last argument, single_integral, is optional and its value must either be
TRUE or FALSE only. It is set to TRUE for single integrations, and FALSE for
multiple integrations (e.g., double or triple integrations).
A single integration means that a function of a single variable, such as func(x),
is integrated:
upper
output 

func( x)dx
lower
A multiple integration means that a function of two or more variables, such as func(x,
y) or func(x, y ,z), is integrated. Our example here, Eq. (1.2), involves a only single
integration, so the single_integral must be set to TRUE. If this argument is left
blank (as we have done here), the value TRUE is assumed. We will later consider
another example in the next section (1.3.5) involving double integrations.
To integrate Eq. (1.2), ITG’s first argument x (cell L2) is the leaf area index, L
(cell E3); the argument func (cell M2) is the leaf photosynthetic function (cell E6),
the argument output (cell N2) is the integration result (canopy photosynthesis) to be
stored in cell E7, and the integration limits: lower (cell O2) and upper (cell P2) are 0
and LAI (cell E2), respectively (Fig. 1.9).
Before we can run the model, we need to specify which variables are to be
saved and where they should be displayed. For this purpose, we need to define two
names, _read and _write, to cells G3 and G6, respectively, and enter the following:
Row
1
2
3
4
5
6
Col. G
to output
LAI
=LAI
Col. H
Cell name
AT
=E7
G3: _read
model output
G6: _write
Cells G3 and H3 instruct BuildIt to save L and AT, respectively. At the end of each
iteration cycle in the inner loop, BuildIt saves these values starting from cells G6 and
H6 onwards.
Choose Start Simulation from the BuildIt menu to start the simulation. The
results are displayed in cells G6 to H26 (Fig. 1.10). From this output list, we can plot
the chart which shows an increase at a decreasing rate in the canopy photosynthesis
with increasing LAI.
22
Fig. 1.10. Model for canopy photosynthesis: its setup and output
1.3.5 Example 3: Daily canopy photosynthesis
In the previous example, Eq. (1.1), the leaf photosynthesis function, AL, was
integrated over the entire canopy, [0, LAI], to give the canopy photosynthesis
function, AT or Eq. (1.2). This equation, however, only gives the canopy
photosynthesis at an instantaneous moment. To determine the daily canopy
photosynthesis (that is, the canopy photosynthesis for the entire day), we must
integrate Eq. (1.2) over the whole day, which is equivalent to integrating the function
from the time of sunrise to sunset (since there is no photosynthesis for periods before
sunrise and after sunset). The daily canopy photosynthesis function, ATd, is as shown
below, expressed in units g CO2 day-1:
ATd 
3600  LAI
1000000
tss LAI
 
tsr 0
Am  kI 0  t  exp  kL 
Am   kI 0  t  exp  kL 
dL dt
(1.3)
where the division by 1000000 is to convert g CO2 to g CO2, multiplication by 3600
and is to convert t, tsr and tss from unit hours to seconds, and further multiplication by
LAI is to convert the photosynthetic rate per unit leaf area to the photosynthetic rate
for the whole canopy. In Eq. (1.3), I0(t) is the solar irradiance (J m-2 ground area s-1) at
time t (hours), given by France and Thornley (1984) as
I0 t  
   t  tsr  
sin 2 

1800  tss  tsr 
  tss  tsr  
I t ,d
tsr  t  tss
23
(1.4)
where It,d is the daily total solar irradiance (J m-2 ground area day-1); and tsr and tss are
the times of sunrise and sunset, respectively (hours). In the previous two examples,
we assumed that the solar irradiance was constant. In this example, however, solar
irradiance is a function of time, t, where I0 varies sinusoidally with time.
We now wish to determine the relationship between ATd and LAI from LAI = 0
to 10 with a 0.5 interval. We will take the same values for the equation variables as
before, with the additional of some new values: let us take It,d as 10 MJ m-2 day-1, tsr
as 6:00 hours, and tss as 18:00 hours.
Create a new workbook and save it as Ex3. Rename one of the worksheets as
DailyCanopyPhotosyn. As usual, we begin by entering the Control section in this
worksheet:
Row
1
2
3
4
5
Col. A
CONTROL
max steps
step size
step
criteria
Col. B
Cell name
10
0.5
0
=_step<=B2
B3: _stepsize
B4: _step
B5: _criteria
The equation variables are entered as:
Row
7
8
9
10
11
12
13
Col A
PARAMETERS
Am
k
e
Itd
tsr
tss
Col. B
Cell name
1500
0.5
12
10000000
6
18
B8: Am
B9: k
B10: e
B11: Itd
B12: tsr
B13: tss
Cells B8 to B10 are for the same variables as before, whereas the new variables are at
B11 (daily total solar irradiance, expressed in J m-2 day-1), B12 (hour of sunrise) and
B13 (hour of sunset).
The section of the worksheet that implements Eq. (1.3) and (1.4) is as follows:
Row
1
2
3
4
5
6
7
8
9
10
11
12
13
14
Col. D
MODEL
Solar
radiation
hour
n1
n2
Io
Leaf photosyn.
LAI
L
n1
n2
AL
Col. E
Cell name
E3: hour
=Itd/(1800*(tss-tsr))
=SIN(PI()*(hour-tsr) /
(tss-tsr))^2
=E4*E5
=_step
E6: Io
E9: LAI
E10: L
=Am*e*k*Io*EXP(-k*L)
=Am+e*k*Io*EXP(-k*L)
=E11/E12
24
Daily canopy
photosyn.
itg out
ATd
15
16
17
=(E16*LAI*3600)/1000000
Cells E3 is the hour, t, and together with cells E4 and E5, they are used to calculate
I0(t) in cell E6. Cells E9 to E13 are to calculate the leaf photosynthetic rate, AL. To
perform a double integration on AL, we require two ITG statements:
Row
2
3
Col. J
canopy
photosyn.
daily canopy
photosyn.
Col. K
itg
Col. L
=L
Col. M
=E13
Col. N
=E16
Col. O
=0
Col. P
=LAI
itg
=hour
=E13
=E16
=tsr
=tss
Col. Q
FALSE
Also define the name _operation to cell K2 to indicate to BuildIt that the list of
operations begin here.
Recall that the ITG operation has the following specification:
ITG (x, func, output, lower, upper, single_integral)
Cells K2 to Q2 specify that the function AL of variable L is to be integrated over the
interval [0, LAI], and the result of the integration is to be stored in cell E16. The last
argument, single_integral, is set to FALSE because the integration of function AL
is still incomplete, and BuildIt should check the next line for further instructions on
integrating this function.
Consequently, cells K3 to Q3 further specify that the function AL of variable
hour is to be integrated over the interval [tsr, tss], and the result of the integration is to
be stored in cell E16 as well. This time, the argument single_integral is left blank
(assumed TRUE by default) because the integration of function AL is now complete.
Both these ITG specifications instruct BuildIt to perform a double integration
on the function AL: once over the whole canopy, then over the whole day. Their
operations can be summarized as
 LAI

output     AL  L, t  dL dt
 0
tsr 

tss
where the result of the double integration, output, is stored in cell E16. In cell E17, the
formula is
=(E16*LAI*3600)/1000000
so that the integration result (cell E16) is expressed in units g CO2 day-1 to finally give
the daily canopy photosynthesis, ATd.
The following setups the model output:
Row
1
2
3
4
5
6
Col. G
to output
LAI
=LAI
Col. H
Cell name
Atd
=E17
G3: _read
model output
G6: _write
25
where _read and _write are defined to cells G3 and G6, respectively.
Choose Start Simulation from the BuildIt menu to start the simulation. The
results are displayed in cells G6 to H26 (Fig. 1.11). From this output list, we can plot
the chart which shows a near linear increase in the daily canopy photosynthesis with
increasing LAI.
Fig. 1.11. Model for daily canopy photosynthesis: its setup and output
1.3.6 Example 4: Prey-predator model
We continue to explore BuildIt’s functionality by building the classical LotkaVolterra’s prey-predator model in Excel. In an ecosystem, there are two species: a
prey and a predator. Their population are closely related to one another. For example,
a large population of prey means more food is available to the predator. This
encourages predation and this will increase the population of predator. Eventually
predation outstrips food supply and causes a decline in the predator population. A low
population of predator now encourages the rise again in the prey population. This
dynamics continue in a cycle of rise and fall in both the prey and predator population.
The rate of population growth over time for the prey and predator can be
described as:
26
dx
  x   xy
dt
(1.5)
dy
  xy   y
dt
(1.6)
where t is the time; x and y are the population for the prey and predator, respectively;
 and  are the growth rates of the prey and predator, respectively; and  and  are the
death rates of the prey and predator, respectively. The prey and predator population at
any one time can be determined, using Euler’s method, as
xt 1  xt 
dx
t
dt
(1.7)
yt 1  yt 
dy
t
dt
(1.8)
where subscripts t and t+1 indicate the population at time step t and the subsequent
time step t+1, respectively; and t is the interval between both time steps t and t+1.
We will simulate the prey and predator population dynamics for 200 time
steps, with a time step interval of 1. So let us begin by fixing the initial values for x
and y as 50 and 15, respectively;  and  as 0.1 and 0.001, respectively; and  and 
as 0.01 and 0.05, respectively.
Create a new workbook and save it as Ex4. Ensure that this workbook has at
least two worksheets. We will separate the model implementation and model output
results by placing them in separate worksheets. This option is entirely your choice:
you may still place both parts together in one worksheet as we have done in the
previous examples. Here, we will place the model implementation in a worksheet
named Model, and the model output results in a worksheet named Output.
Recall that we wish to run our prey-predator model for 200 time steps, with a
time step interval of 1, so in the Model worksheet, enter the following for the Control
section:
Row
1
2
3
4
5
6
Col. A
CONTROL
Max steps
Step size
Step
Criteria
No.
subintervals
Col. B
Cell name
200
1
0
=(_step/_stepsize)<B2
2
B3: _stepsize
B4: _step
B5: _criteria
What is crucially different in the Control section as compared to the previous
examples is cell B6. It denotes the number of subintervals within a single time step.
The purpose of dividing the interval of a single time step into, preferably, two or more
subintervals is to reduce the error of prediction.
Let us consider just the prey population, x. Fig. 1.12 shows that at time step t,
the prey population is x(t), and the calculated population growth rate is dx(t)/dt. So in
the next time step t+1, the prey population, calculated using Eq. (1.7), increases to
x(t+1) (indicated by point A). Point A, however, assumes that within the time interval,
t, the population growth rate remained constant, but Fig. 1.12 shows that this is
27
untrue. The rate of population growth actually decreased within this period, so the
actual prey population is really at point B. If t is large, the risk increases that the
predicted value may deviate far from the true value (i.e., distance between A and B is
large).
Fig. 1.12. The error in predicting x(t+1) can be large when the growth rate, dx(t)/dt, is
assumed constant over a large time interval, t
Consequently, to reduce the prediction error, we should divide the time
interval into two or more subintervals so that for each subinterval we calculate the
growth rates and subsequent population for both prey and predator (Fig. 1.13).
Because we have two or more subintervals within t, Eq. (1.7) and (1.8) are modified
to
xt 1  xt 
t  n 1 dxt  i n 


n  i  0 dt 


(1.9)
t  n 1 dyt  i n 

yt 1  yt   
n  i  0 dt 


(1.10)
where n is the number of subintervals. The more subintervals we have, the more
accurate our estimates are, but the trade off is calculations become more intensive and
time-consuming which can be a significant problem in large, complex models. How to
choose the value for n often requires some trial-and-error which involves examining
the gain in prediction accuracy with increasing n. We may opt for a particular value
for n when prediction accuracy can no longer be increased significantly or when
simulation runs become too lengthy if a higher n is used. Generally, models with large
time step intervals or with sensitive rates of change require a large n.
In this example, the number of subintervals is set at two (cell B6). Increasing n
to more than two did not improve the accuracy of predictions significantly, so we will
maintain n as 2.
28
Fig. 1.13. The error in predicting x(t+1) reduces when the time interval, t, is divided
into two (or more) subintervals
We can now proceed to enter the model parameters in the Model worksheet:
Row
8
9
10
11
12
13
Col. A
PARAMETERS
Population
prey
predator
Growth rate
prey
Col. B
14
predator
0.001
15
16
Death rate
prey
0.01
17
predator
0.05
Cell name
50
15
0.1
B13:
prey_growth
B14:
pred_growth
B16:
prey_death
B17:
pred_death
For the prey, its initial population is in cell B10, growth rate in cell B13, and death
rate in cell B16. For the predator, its initial population is in cell B11, growth rate in
cell B14, and death rate in cell B17. Cells B13, B14, B16 and B17 are further defined
with their respective cell names.
The following entries implements in the model in the Model worksheet:
Row
1
2
3
Col. D
MODEL
4
Predator
Prey
Col. E
Col. F
Col. G
Cell name
pop (t1)
rate
=prey_growth*
prey_pop
- prey_death
* prey_pop
* pred_pop
=pred_growth*
prey_pop
* pred_pop
pop (t2)
=prey_pop+F3*
_stepsize/B6
E3: prey_pop
29
=pred_pop+F4*
_stepsize/B6
E4: pred_pop
- pred_death
* pred_pop
Notice, however, that both cells E3 and E4 are blanks. This is because we will
instruct BuildIt to initialize their values using the INI operation which has the
following specification:
INI (destination, source)
INI
specifies that the contents from source cells will be copied to the cells referred to
in the destination cells. In addition, this initialization operation must be specified in
the Prerun section so that before BuildIt enters the inner loop, the prey and predator
population have been initialized to their given values.
So in the Model worksheet, enter the following for the Prerun section:
Row
8
9
10
Col. I
PRERUN
Initial prey pop.
Initial predator pop.
Col. J
Col. K
Col. L
ini
ini
=prey_pop
=pred_pop
=B10
=B11
Ensure that cell J9 is defined with the name _prerun to indicate to BuildIt that this
cell is the starting point for all prerun operations. In this example, cells B10 and B11
are the source cells, and cells E3 (prey_pop) and E4 (pred_pop) the destination cells.
The first INI operation (cells J9 to L9) instructs BuildIt to copy the contents of cell
B10 to E3, initializing the prey population. The second INI operation (cells J10 to
L10) copies cell B11 to E4, initializing the predator population. Note that cells I8, I9
and I10 are labels for documentation purposes only.
As discussed earlier, we wish to reduce the prediction errors by dividing the
time step interval into two or more subintervals so that we can determine the prey and
predator population at each successive subinterval. We earlier fixed the number of
subintervals, n, to two (cell B6). So while cell E3 is the prey population at time step t,
or xt, subsequent calculations make cell G3 the prey population at time step t+0.5, or
xt+0.5. We like to somehow copy the value in cell G3 to E3 so that cell E3 is now xt+0.5,
and subsequent calculations, as before, yields xt+1 in cell G3. Since we have two
subintervals, we should copy cell G3 to E3 again. This gives cell E3 the value of xt+1.
In other words, prior to the REP operation, cell E3 holds xt, and after the REP
operation, cell E3 holds xt+1. The same development likewise occurs in cells E4 to G4
to calculate the predator population.
To instruct the copying of source cells to destinations cells, we use BuildIt’s
operation which has the following specification:
REP
REP (destination, source, num_of_replacements)
which specifies that the contents from source cells will be copied to the cells referred
to in the destination cells, and this copying operation will be done
num_of_replacements number of times. So in the Model worksheet, enter the
following:
Row
1
2
Col. I
OPERATION
update population
Col. J
Col. K
Col. L
Col. M
rep
=E3:E4
=G3:G4
=B6
Ensure that cell J2 is defined with the name _operation to indicate to BuildIt the list
of operations in the Operation section that must be executed in every iteration in the
30
inner loop. Here, cells E3:E4 are the destination cells and cells G3:G4 the source
cells, and their copy operation occurs twice (cell B6).
We have completed implementing the model in the Model worksheet. We can
now specify the model output. Fig. 1.14 shows the Output worksheet, where we
entered the following to specify that we wish to output the time step, prey population
and predator population:
Row
1
2
3
4
5
6
7
Col. A
to output
step
=_step
Col. B
Col. C
Cell name
prey
=prey_pop
FALSE
predator
=pred_pop
FALSE
A3: _read
model output
A7: _write
Cell A3 is defined with the name _read. This tells BuildIt that this cell and all cells to
its right are variables to be saved. Recall that BuildIt actually checks two rows: the
first row is A3 to C3, the variables to be saved, and the second row is to instruct
BuildIt whether to either save the corresponding variables before or after the
executions of the operations in the Operation section (Fig. 1.5).
Fig. 1.14. Output listing in the Output worksheet for the prey-predator model
Cell A4 is left blank (assumed TRUE by default) so _step is saved after the
operations have been executed in the inner loop. Cells B4 and B5 are instead FALSE
so prey_pop and pred_pop variables are saved before these operations are executed.
As mentioned previously, the value in cell E3 (prey population) is different
prior and after the REP operation. Consequently, we want to save the prey population
before the REP operation is done (i.e., by setting cell B4 to FALSE). If we save the
31
prey population after the REP operation (i.e., by setting cell B4 to TRUE or leaving it
blank), we will get a mismatch in the output sequence, where at time step t, for
instance, the corresponding prey population saved is actually xt+1 (xt is not saved at
all) (Fig. 1.15).
Fig. 1.15. Mismatch in the output sequence due to setting the output for the prey and
predator populations to TRUE (cells B4 and C4). Compare this output to Fig. 1.14.
For some variables, it does not matter if we set TRUE or FALSE in their
corresponding cells in the second row. The variable _step, for example, can be saved
before or after the Operation section because the value of _step is not changed by the
operations.
To start the simulation, choose Start Simulation from the BuildIt menu. Fig.
1.14 shows a part of the model output list. Cell A7 is defined with the name _write
so output begins here and spreads until to cell C206 (not shown). The output list is
then used to plot a chart showing the dynamics of the prey and predator population.
We see that the prey population shows greater fluctuations than the predator
population. The prey population fluctuates between about 21 to 97, whereas the
predator population fluctuates in a narrower range of about 6 to 16. Also note that the
predator population is often below the initial population of 15, whereas the prey
population can at times exceed the initial population of 50.
Before we consider the next example, it is important to know that there may
be a difference between having a single and multiple REP operations to copy a range
of cells to their respective destinations. For instance, we earlier entered the following
for the Operation section:
Row
1
2
Col. I
OPERATION
update population
Col. J
Col. K
Col. L
Col. M
rep
=E3:E4
=G3:G4
=B6
32
This case copies cells G3 to E3 and G4 to E4 simultaneously. But what if we split the
single REP operation into two: one operation to copy cell G3 to E3, and another for G4
to E4? That is, we enter the following:
Row
1
2
3
Col. I
OPERATION
update prey population
update predator population
Col. J
Col. K
Col. L
Col. M
rep
rep
=E3
=E4
=G3
=G4
=B6
=B6
This second case has two REP operations: one for each copy operation.
The difference between these two cases is subtle. Eq. (1.5) and (1.6) reveal
that prey and predator population depend on each other; that is, cell G3 depends on
E4, and G4 on E3. This interdependency means copying cell G3 to E3 and G4 to E4
must occur at the same time. In the second case, the first REP operation copies cell G3
to E3 which updates cell E3 and causes recalculations to determine the new predator
population in cell G4. The old value in cell G4 is overwritten before it could be
copied to cell E4. So by the time the second REP operation is executed, it is this new
(but wrong) value in cell G4 that is copied to cell E4. Cell E4 has been updated with a
new value which causes recalculations to give a new (but wrong) value to cell G3, and
so this error proliferates throughout the simulation run where wrong values are used
in calculations to produce wrong answers.
1.3.7 Example 5: Simple plant growth model
In this fifth example, we will implement a simple plant growth model adapted
from Thornley (1977). Fig. 1.16 shows that the assimilates produced by
photosynthesis is partitioned to maintenance for the plant’s continual survival and to
growth for the synthesis of new materials for the plant. The plant weight consists of
two components: the storage weight and the plant structural weight. The storage
weight, Ws, is supported by the addition of newly produced substrates via the
photosynthesis process. A portion of the plant’s storage, kgWs, will be utilised for
maintenance and growth, where of this total (kgWs), Yg of it will be utilised for growth,
and the remainder (1 – Yg) for maintenance. Of the total plant structural weight, Wg, kd
of it will degrade and the substrates returned to the storage, and ks of it will senesce
(i.e., loss of materials due to increasing age).
The rates of change in the dry weights for storage and structure components at
time t (in unit hours) are thus
dWs
12
 kd Wg  k gWs 
 3600  LAI  AT
dt
44  1000000
dWg
dt
 k gWsYg  kdWg  ksWg
(1.11)
(1.12)
As shown earlier, Eq. (1.2) gives us the canopy photosynthesis, AT, in units of
g CO2 m-2 leaf area s-1 which we will convert to units g C m-2 leaf area s-1 by:
AT ( g CO2 m-2 s-1 )  AT 
12
(g C m-2 s-1 )
44  1000000
Every 44 g of CO2 has 12 g of C since the atomic weight of carbon and molecular
weight of carbon dioxide are 12 and 44 g, respectively. Therefore, AT g CO2 will
have (12/44)AT g of C. We divide AT by 1000000 to convert the unit g C to g C.
33
Since time t is in unit hour, we need to multiply AT by 3600 (i.e., 60 s  60 mins), and
further multiplication by LAI is to convert AT into units g C m-2 ground area hour-1;
hence, the converted AT is now compatible with the dry weight units (which are
expressed as g C per unit ground area)
Fig. 1.16. A simple plant growth model (adapted from Thornley, 1977)
The cumulative increase in the dry weights for storage and structure from time
t1 to t2 are obtained by integrating their rates of change function over the period [t1,
t2]:
Ws 
t2


12
 kd Wg  k gWs  44 1000000  3600  LAI  AT dt
t1
12  3600  LAI
 kd Wg  k g Ws t 
44  1000000


t2
(1.13)
 AT dt
t1
t2
 k gWsYg  kd Wg  ksWg  dt
Wg 
t1

(1.14)

 k g Ws Yg  kd Wg  ksW t
where t is the time interval (t2 - t1), and kg ks and Yg are taken as constants.
Calculations are straightforward with the exception of the integration of the canopy
photosynthesis AT. From Eq. (1.2), canopy photosynthesis is
LAI
AT 

0
LAI
AL dL 

0
Am  kI 0 exp  kL 
Am   kI 0 exp  kL 
dL
34
where solar irradiance, I0, is not a constant but varies with time t according to Eq.
(1.4) as
I0 t  
   t  tsr  
sin 2 

1800  tss  tsr 
  tss  tsr  
I t ,d
tsr  t  tss
where It,d is the daily total solar irradiance (J m-2 ground area day-1); and tsr and tss are
the times of sunrise and sunset, respectively (hour).
In this example, we will determine the daily increment in the total dry weights
of Ws and Wg, with t1 = 0 and t2 = 24. The values for the rest of the parameters are as
listed in Table 1.2. Since there is no photosynthesis for periods before sunrise and
after sunset, integration of canopy photosynthesis over the whole day is equivalent to
its integration over the period between sunrise and sunset so Eq. (1.13) becomes
12  3600  LAI
Ws  kd Wg  k g Ws t 
44  1000000


tss

AT dt
(1.15)
tsr
Consequently, the dry weights for storage and structure can be determined by:
Ws,t 1  Ws,t  Ws
(1.16)
Wg ,t 1  Wg , t  Wg
(1.17)
where subscripts t and t+1 denote the dry weight at time t and the subsequent time
t+1, respectively.
Table 1.2. Parameter values for the simple plant growth model
Symbol Description
Units
Value
Am
max. leaf photosynthesis rate
g CO2 m-2 leaf area s-1
1500

solar radiation conversion factor g CO2 J-1
12
k
canopy extinction coefficient
-
0.5
LAI
leaf area index
m2 leaf area m-2 ground area 2
kg
growth rate constant
day-1
1.98
kd
degradation rate constant
day-1
0.10
ks
senescence rate constant
day-1
0.05
Yg
efficiency of structural synthesis -
Ws
substrates dry weight
g C m-2 ground area
20
Wg
structural dry weight
g C m-2 ground area
130
tsr
time of sunrise
hour
6
tss
time of sunset
hour
18
It,d
daily total solar irradiance
J m-2 ground area day-1
10000000
0.75
Let us now implement Thornley’s simple plant growth model in Excel. Create
a workbook and name it as Ex5. Rename one of the worksheets as CropGrowth,
where in this worksheet, enter the following for the Control section:
35
Row
1
2
3
4
5
Col. A
CONTROL
max steps
step size
step
criteria
Col. B
Cell name
10
1
0
=(_step/_stepsize)<B2
B3: _stepsize
B4: _step
B5: _criteria
Cell B3 indicates that the time interval is 1 day, and cells B2 and B5 indicate that we
will run the simulation for 10 days.
The next is to enter the model parameters (listed in Table 1.2) as follows:
Row
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
Col. A
PARAMETERS
Am
k
quantum
LAI
kg
kd
ks
Yg
tsr
tss
Itd
INITIAL
Ws
Wg
Col. B
Cell name
1500
0.5
12
2
1.98
0.1
0.05
0.75
6
18
10000000
B8: Am
B9: k
B10: e
B11: LAI
B12: kg
B13: kd
B14: ks
B15: Yg
B16: tsr
B17: tss
B18: Itd
20
130
The implementation of the calculations for solar irradiance and daily canopy
photosynthesis are as follows:
Row
1
2
3
4
Col. D
MODEL
Solar
radiation
hour
n1
5
n2
6
7
8
Io
9
10
Leaf
photosyn.
L
n1
11
n2
Col. E
Col. F
Col. G
Cell name
E3: hour
=Itd /
(1800 *
(tss-tsr))
=SIN(PI()*
(hour-tsr)
/(tsstsr))^2
=E4*E5
E6: Io
E9: L
=Am*e*k*Io*
EXP(-k*L)
=Am+e*k*Io*
EXP(-k*L)
36
12
13
14
15
16
17
18
19
20
21
AL
=E10/E11
Daily
canopy
photosyn.
itg out
ATd
=(12/44)*36
00*LAI*
E15/1000000
E16: ATd
Weight
wgt(t1)
storage
structure
wgt. gain
=(kd*Wgkg*Ws)*
_stepsize+
ATd
=(kg*Ws*Yg
-kd*Wgks*Wg)*
_stepsize
wgt(t2)
=Ws+F20
E20: Ws
=Wg+F21
E21: Wg
Cells D1 to G21 contain the various calculations for determining the solar
irradiance and daily canopy photosynthesis. Cell E6 contains the solar irradiance for
the hour as calculated by Eq. (1.4).
BuildIt’s operation, ITG, is used to integrate Eq. (1.1) over the whole canopy
and whole day to determine the daily canopy photosynthesis. Once that is determined,
the weight gain by the plant’s storage and structure components are determined using
BuildIt’s REP operation. Both ITG and REP operations are entered in the Operations
section as follows:
Row
1
2
3
Col. Q
canopy
photosyn.
daily
canopy
photosyn.
update
wgt.
Col. R
itg
Col. S
=L
Col. T
=E12
Col. U
=E15
Col. V
0
Col. W
=LAI
itg
=hour
=E12
=E15
=tsr
=tss
rep
=E20:
E21
=G20:
G21
Col. X
FALSE
Cell R1 is defined with the name _operation to indicate the start of the Operation
section. In this section, there are three operations: two ITG and one REP. Cells Q1 to
Q3 are merely labels to describe their respective operations. As described earlier in
section 1.3.5, the first two ITG operations are to determine the daily canopy
photosynthesis which involves double integration of the function AL or Eq. (1.1): first
over the whole canopy (cells R1 to X1), then over the day from sunrise to sunset
(cells R2 to W2). The result, the daily canopy photosynthesis, is stored in cell E12,
after which the dry weights of storage (cell G20) and structure (cell G21) components
can be determined for the next time step, t+1. Their values are then copied to cells
E20 and E21, respectively (using the REP operation in cells R3 to T3), and the whole
cycle repeats.
Note that in the Operation section, the REP is entered as follows:
Row
3
Col. Q
update
wgt.
Col. R
rep
Col. S
=E20:
E21
Col. T
=G20:
G21
37
Col. U
Col. V
Col. W
Col. X
The single REP operation above is not equivalent to having two separate REP
operations: one for copying cell G20 to E20, and another for G21 to E21; that is,
Row
3
4
Col. Q
update
storage
wgt.
update
structure
wgt.
Col. R
rep
Col. S
=E20
Col. T
=G20
rep
=E21
=G21
Col. U
Col. V
Col. W
Col. X
These two cases are not the same because the dry weight Ws,t+1 (cell G20)
depends on Wg,t (cell E21), and Wg,t+1 (cell G21) on Ws,t (cell E20). Therefore, their
copy operations must occur at the same time because their values depend on each
other. Separate copy operations cause premature calculations which give wrong
values which will then be copied to other locations, proliferating the error (see also
section 1.3.6).
A Prerun section is required to initialize the dry weights (cells E20 and E21).
So enter the following:
Row
7
8
9
Col. Q
PRERUN
initial storage wgt.
initial structural wgt.
Col. R
Col. S
Col. T
ini
ini
=Ws
=Wg
=B21
=B22
Ensure that cell R8 is defined with the name _prerun so that BuildIt will examine
cells R8 to T9 to initialise the storage dry weight (cell E20) with cell B21 and
structure dry weight (cell E21) with cell B22.
For model output, we will choose to output the day, daily canopy
photosynthesis, dry weight gains in storage and structure, individual dry weight for
storage and structure, and their total dry weight (Fig. 1.17). The entries are as follows:
Row
1
2
Col. I
to output
step
Col. J
Col. K
Col. L
Col. M
Col. N
Col. O
ATd
gain Ws
gain Wg
Ws
Wg
3
4
5
6
=_step
=ATd
=F20
FALSE
=F21
FALSE
=Ws
FALSE
=Wg
FALSE
total
wgt.
=M3+N3
FALSE
model
output
7
Cell I3 and I7 are defined with the names _read and _write, respectively. Note that
cells K4 to O4 are FALSE to indicate that their cell contents in the corresponding row
above should be saved before the ITG and REP operations are executed. Also note that
cell O3 sums Ws and Wg, giving their total weight which will be saved in every
iteration cycle.
Fig. 1.17 shows the model output results. It shows a linear-like increase in the
total plant dry weight and the structure dry weight (Wg). The dry weight for storage
remained nearly constant. Because we fixed the daily solar irradiance and the times
for sunrise and sunset, the daily canopy photosynthesis (ATd) is always the same
throughout the simulation period.
38
Fig. 1.17. Model output listing for the simple plant growth model
1.3.8 Example 6: Factorial and triangular numbers
This example demonstrates BuildIt’s ACC operation which determines either
the summation or product of a series of numbers. The ACC operation has the following
specification:
ACC (x, accumulated_x, operator)
where operator can either be + or * for addition or multiplication, respectively. If the
operator is addition (+), ACC determines the summation of a series of numbers; that
is,
accumulated_x 
 xi
i
But if the operator is multiplication (*), ACC determines the product of a series of
numbers; that is,
accumulated_x 
 xi
i
The factorial of a natural number n, expressed as n!, is determined by
39
n! 
n
x
n0
(1.18)
x 1
so that, for instance, 5! = 54321 = 120 and 3! = 321 = 6. On the hand, Tn is a
triangular number which is the sum of n natural numbers from 1 to n, and it is
determined by
Tn 
n
x
n0
(1.19)
x 1
Rather than summing every number from 1 to n, Tn can be determined alternatively by
the formula: n(n+1)/2. For demonstration purposes, however, we will ignore this
faster and easier formula to determine Tn.
We will implement in Excel both Eq. (1.18) and (1.19) to determine the
factorial and triangular number of any given natural number n. Create a workbook and
save it as Ex6. Rename one of the worksheet as Model. Enter the following in the
Model worksheet:
Row
1
2
3
4
5
6
7
Col. A
CONTROL
Step size
Step
Criteria
VALUE
n
Col. B
Cell name
1
0
=x<=n
B2: _stepsize
B3: _step
B4: _criteria
5
B7: n
and for the implementing the model:
Row
1
2
3
4
5
6
7
8
9
Col. D
MODEL
x1
Col. E
Cell name
x2(=x1+1)
=x+1
D3: x
Product
Sum
Factorial
Triangular
=IF(B7>0,E5,1)
=IF(B7>0,E6,#N/A)
Note that, unlike previous examples, there is no use for the maximum number
of steps to terminate the inner loop iterations. Instead, the iterations will end only
when the value of x (cell D3) is more than n (cell B7). In other words, iterations end
when x<=n is FALSE (cell B4).
Cell E5 holds the factorial of n, whereas cell E6 the triangular number of n.
However, cell E8 handles the unique case for n = 0, so that 0!, by definition, is equal
to 1. Likewise, cell E9 handles the case for T0 which returns the error message #N/A.
Note that this worksheet assumes that n is always zero or positive.
We need to use both of BuildIt’s ACC and REP operations to implement
Eq.(1.18) and (1.19). The following shows how:
40
Row
1
2
3
4
Col. G
OPERATION
factorial
triangular
copy x2 to x1
Col. H
Col. I
Col. J
Col. K
acc
acc
rep
=D3
=D3
=D3
=E5
=E6
=E3
*
+
Ensure that the name _operation is defined for cell H2 to specify to BuildIt that the
list of operations begin from this cell.
The first row, cells J2 to M2, instructs BuildIt to multiply x (cell D3) to the
current product in cell E5, whereas the second row, cells J3 to M3, instructs BuildIt to
add x (cell D3) to the current sum in cell E6. Both cells E5 and E6 will be updated by
BuildIt to reflect their new summation and product values, respectively. After these
two operations, cells J4 to M4 instruct BuildIt to copy the value from cell E3 to D3.
Recall that the formula in cell E3 is x+1, so every iteration cycle increments x by one,
as required by Eq. (1.18) and (1.19). Note that cell M3 is left blank, so the default
value 1 is assumed for the num_of_replacements argument for the REP operation.
A Prerun section is required to initialize the x variable (cell D3). So enter the
following:
Row
6
7
Col. G
PRERUN
initial x1=1
Col. H
Col. I
Col. J
ini
=x
1
Ensure that cell H7 is defined with the name _prerun so that BuildIt will examine
cell H3 to J3 to initialise x (cell D3) with the value 1 (cell J7).
Unlike previous examples, there is no need to define the names _read and
_write because cells E8 and E9 will contain the factorial and triangular number of n,
respectively.
Choose Start Simulation from the BuildIt menu to start the simulation, and
Fig. 1.18 shows that 5! and T5 are 120 and 15, respectively. You can change cell B7 to
a different value for n and determine its factorial and triangular number by running
the model again.
Fig. 1.18. The setup and result for the factorial and triangular number of n
1.3.9 Example 7: Scenarios
In simulation, we are often interested to determine the predicted outcome of
different scenarios. This is often done by changing the values of some of the model
parameters and running the simulation again to obtain the predicted outcome. In
41
Example 4 (section 1.3.6), we saw the dynamics of population rise and fall for both
prey and predator (Fig. 1.14). What if we double the growth rates for both the prey
and predator? That is, what happens to the population dynamics when we change the
growth rate for prey from 0.1 to 0.2, and for predator from 0.001 to 0.002?
One way is to change the value in cells B13 and B14 (in the Model worksheet
of the Ex4 workbook) to 0.2 and 0.002, respectively. We then run the model again,
and compare its output with the original outcome (with lower growth rates).
Alternatively, we can use BuildIt’s functionality for running scenarios. In the Ex4
workbook, enter the following in the Model worksheet:
Row
15
16
Col. I
SCENARIO
2x growth rates
Col. J
Col. K
Col. L
Col. M
=prey_growth
0.2
=pred_growth
0.002
We also define the name _scenario to cell J16 to indicate to BuildIt the start of the
Scenario section and where our scenario list is located. Cell I15 and I16 are optional
and serve as labels for the scenario operation.
Whenever we define a cell with the name _scenario, BuildIt will begin
looking at pairs of cells. The search starts with the cell defined with _scenario and
moves progressively right, examining each cell pairs. BuildIt stops searching only
when it encounters a blank cell. We defined _scenario to cell J16 in our example, so
BuildIt will look at the pair of cells (J14, K14), followed by (L14, M14). BuildIt stops
at (L14, M14) because the cell N14 (the cell right of M14) is blank (not shown).
BuildIt treats each cell pair as follows:
(destination, source)
where the contents from the source cell is copied to that specified in the
destination cell.
To better explain this, let us now take the first cell pair, (J14, K14), which has
the following formulas: (prey_growth, 0.2). This indicates to BuildIt to give
prey_growth (cell B13) a value of 0.2. The second cell pair, (L14, M14), has the
following formulas: (pred_growth, 0.002) which indicates to BuildIt that
pred_growth (cell B14) will have the value 0.002.
When we run the model, BuildIt first runs the simulation for the original
growth rates of 0.1 and 0.001 for the prey and predator, respectively. Once the
iterations in the inner loop stop, BuildIt checks for any more given scenarios. Since
we have specified one more, iterations in the inner loop restart using the doubled
growth rates for both prey and predator. BuildIt repeats this process for subsequent
scenarios.
To demonstrate this process, let us create three more scenarios. In other words,
we have four scenarios in total: the first is with the original growth rates for prey and
predator. The second scenario, as stated earlier, is the doubling of their growth rates.
Why not also triple their growth rates as the third scenario? Finally, for the last
scenario, we double both growth and death rates for both prey and predator. For these
additional three scenarios, we implement them like this in the Model worksheet (for
brevity, the labels in column I are not shown):
Row
16
Col. J
=prey_
growth
Col. K
0.2
Col. L
=pred_
growth
Col. M
0.002
42
Col. N
Col. O
Col. P
Col. Q
17
18
=prey_
growth
=prey_
growth
0.3
0.2
=pred_
growth
=pred_
growth
0.003
0.002
=prey_
death
0.2
=pred_
death
0.1
As before, ensure that cell J16 has been defined with the name _scenario. Cells J16
to M16 form the second scenario (doubling growth rates for prey and predator). The
next row, cells J17 to M17, is the third scenario (tripling growth rates for prey and
predator). The fourth and last scenario is specified from cell J18 to Q18 (doubling all
rates for prey and predator).
Consequently, we will have a set of four model output: one for the original
scenario (that is, as shown partially by Fig. 1.14), and the next three are for the
additional scenarios. In the output listing, each output set is separated by a blank row,
and a row header (Fig. 1.19). We see that doubling the growth rates for both prey and
predator changes the dynamics of their population significantly. As compared to the
original growth rate, doubling the growth rate for prey did not cause a larger growth
in its population. Instead, it showed a smaller increase in their population growth as
the prey was suppressed by the larger population growth for the predator. Before, the
predator population seldom exceeded their initial population of 15. But by the
doubling their growth rate, predator population fluctuated more, between about 8 to
37. Similar comparisons can be done for the other two scenarios (not shown here).
Fig. 1.19. Running scenarios for the prey-predator model
Specifying scenarios does not have to be limited to replacing a value with
another value. You could replace a value or formula with another formula. For
instance, In the Ex1 workbook, cell E5 in the LeafPhotosyn worksheet gives the leaf
photosynthetic rate, AL or Eq. (1.1). To keep this example simple, let us say that we
43
wish to take half its calculated value as one of the modeling scenarios. So in the
LeafPhotosyn worksheet in the Ex1 workbook, enter the following:
Row
1
2
Col. J
SCENARIO
half AL
Col. K
Col. L
=E5
=0.5*E3/E4
As usual, we have to define the name _scenario and it is assigned to cell K2. Note
that cell L2 (source) is not a value, but a formula that will be copied to cell E5
(destination), giving half AL at all times. Recall that cells E3 and E4 are the numerator
and denominator in Eq. (1.1), respectively.
We have to be careful, however, that we do not write formulas that will refer
to itself after the copy operation. For instance, consider the following:
Row
1
2
Col. J
SCENARIO
half AL
Col. K
Col. L
=E5
=0.5*E5
The formula in cell L2, =0.5*E5, will be copied to cell E5 after which the formula in
cell E5 will now read as
=0.5*E5
which refers to itself. This self-referencing (or circular) formula causes subtle errors
which may not be easily detected.
44
Chapter 2. Calculations in the generic crop growth model
2.1
Meteorology
This chapter documents all the equations used in the gcg (generic crop growth)
model. Calculations begin with the meteorology section. Properties such as solar
irradiance, solar position, wind speed, air temperature and vapor pressure must be
known at the outset before calculating other factors that affect plant growth and yield.
2.1.1 Daily properties
Daily properties refer to meteorological properties for a given day of year (or
date).
2.1.1.1 Solar declination
  0.4093cos  2  td  10  365
(2.1)
where  is the solar declination (radians); and td is the day of year.
2.1.1.2 Sunrise, sunset and daylength
 sin   sin  
acos  
  24  tss

 cos   cos  
(2.2)
 sin   sin  
acos  


 cos   cos  
(2.3)
tsr  12 
12
tss  12 
12
DL 
 sin   sin  
acos  
  tss  tsr

 cos   cos  
24
(2.4)
where tsr, tss and DL are the sunrise, sunset and daylength, respectively (all expressed
in hours);  is the solar declination (radians); and  is the site latitude (radians).
2.1.1.3 Solar constant


Ic  1370  1  0.033cos  2  td  10  / 365
(2.5)
where Ic is the solar constant (W m-2); and td is the day of year.
2.1.1.4 Daily extraterrestrial solar irradiance

2
Iet , d  3600  Ic  24  a  acos   a b   b 1   a b  


a  sin  sin  and b  cos  cos 
(2.6)
where Iet,d is the daily extraterrestrial solar irradiance (J m-2 day-1); Ic is the solar
constant (W m-2);  is the solar declination (radians); and  is the site latitude
(radians).
2.1.1.5 Daily total solar irradiance
s 

It , d  I et , d  b0  b1

DL 

(2.7)
45
where It,d is the daily total solar irradiance (J m-2 day-1); Iet,d is the extraterrestrial solar
irradiance (J m-2 day-1); b0 and b1 are empirical coefficients; s is the duration of
sunshine hours (hours); and DL is the day length (hours).
Note: The Angstrom coefficients b0 and b1 used for calculating daily solar radiation
for different climate zones
Climate zones
Cold or temperate
Dry tropical
Humid tropical
b0
0.18
0.25
0.29
b1
0.55
0.45
0.42
Direct and diffuse components
It , d I et , d  0.07
 It , d

2
 I 1  2.3 I
t , d I et , d  0.07  0.07  It , d I et , d  0.35
 t , d 

I df , d  
 I 1.33  1.46 I
0.35  It , d Iet , d  0.75
t , d I et , d 
 t, d 
0.23It , d
0.75  It , d I et , d



I dr , d  It , d  I df , d
(2.8)
(2.9)
where Idf,d is the daily diffuse solar irradiance (J m-2 day-1); Idr,d is the daily direct solar
irradiance (J m-2 day-1); It,d is the daily total solar irradiance (J m-2 day-1); and Iet,d is
the extraterrestrial solar irradiance (J m-2 day-1).
2.1.2 Hourly properties
Hourly properties refer to meteorological properties for a given hour and day
of year (or date and time).
2.1.2.1 Hour angle

   th  12 
12
(2.10)
where  is the hour angle (radians); and th is the local solar time (hours).
Note: hour angles will be negative for before solar noon (th < 12), positive for after
solar noon (th > 12), and zero at exactly solar noon.
2.1.2.2 Solar position
Solar inclination (and solar height)
  acos sin   sin   cos   cos   cos 
(2.11)
where  is the solar inclination (angle from vertical) (radians);  is the solar
declination (radians);  is the hour angle (hours); and  is the site latitude (radians).
Note: Rather than its angle from vertical, solar angle can be specified alternatively as
, the solar height or solar elevation, which is the solar angle from horizontal,
determined by /2 -  (radians).
46
Solar azimuth

 sin   sin   sin 
  acos 

 cos   cos 

  acos  sin   sin   sin 


 cos   cos 


 th  12


 th  12

(2.12)
where  is the solar azimuth (angle from North in a clockwise direction) (radians); 
is the solar declination (radians);  is the hour angle (hours); and  is the site latitude
(radians).
2.1.2.3 Hourly extraterrestrial solar irradiance
Iet  Ic sin   Ic cos 
(2.13)
where Iet is the hourly extraterrestrial solar irradiance (W m-2); Ic is the solar constant
(W m-2);  is the solar inclination (radians); and  is the solar height (or solar
elevation) (radians).
Note:  = /2 -  .
2.1.2.4 Hourly total solar irradiance
It  a    b   cos  th / 12 
 It , d 86400
 
a  acos   a b   b 1   a b 
(2.14)
2
a  sin  sin  and b  cos  cos 
where It is the hourly total solar irradiance (W m-2); It,d is the daily total solar
irradiance (J m-2 day-1); th is the local solar hour (hours); and  is the solar declination
(radians);  is the site latitude (radians).
Direct and diffuse components
It I et  0.22
 It

2
 

 
It

 It 1  6.4 
  0.22  It I et  0.35
I df   
I

0.22
 et
 
 
0.35< It I et  K
 It 1.47  1.66 It I et 

K  It I et
 It  R
(2.15)
R  0.847 – 1.61sin   1.04sin 2 
(2.16)
K  1.47 – R  /1.66
(2.17)
I dr  It  I df
(2.18)
-2
-1
where Idf is the hourly diffuse solar irradiance (J m day ); Idr is the hourly direct
solar irradiance (J m-2 day-1); It is the hourly total solar irradiance (W m-2); Iet is the
47
hourly extraterrestrial solar irradiance (W m-2); and  is the solar height (or solar
elevation) (radians).
Note:  = /2 - , where  is the solar inclination.
2.1.2.5 Hourly air temperature

Tmin  Tset  24  th  tss 
th   t sr  1.5 
Tset 
 tsr  1.5   24  tss 


   th  tsr  1.5  

Ta  Tmin  Tmax  Tmin  sin 
  tsr  1.5   th  t ss (2.19)
tss  tsr




T  Tmin  Tset  th  tss 
th  tss
 set  tsr  1.5    24  tss 
   tss  tsr  1.5  
Tset  Tmin  Tmax  Tmin  sin 

tss  tsr


(2.20)
where Ta is the air temperature (C) at local solar time th (hours); Tmin and Tmax are the
minimum and maximum air temperature (C), respectively; and tsr and tss are the
times of sunrise and sunset, respectively (hours); and Tset is the air temperature (C) at
sunset (tss).
2.1.2.6 Hourly wind speed
Hourly wind speed (m s-1) is assumed uniform throughout the day and equal to the
measured daily wind speed.
2.1.2.7 Saturated air vapor pressure
es Ta   6.1078exp 17.269Ta Ta  237.3 
(2.21)
where es(Ta) is the saturated vapor pressure (mbar) at air temperature Ta (C).
2.1.2.8 Slope of the saturated vapor pressure curve

25029.4exp 17.269Ta Ta  237.3
Ta  237.32
(2.22)
where  is the slope of the saturated vapor pressure curve (mbar K-1 or equivalently,
mbar C-1) at air temperature Ta (C).
2.1.2.9 Hourly air vapor pressure
ea  RH  es (Ta ) 100
(2.23)
where ea is the air vapor pressure (mbar); RH is the relative humidity (%); and es(Ta)
is the saturated air vapor pressure (mbar) at air temperature Ta (C).
Air vapor pressure deficit
Air vapor pressure deficit is simply the difference between saturated and
current air vapor pressure; that is,
48
D  es (Ta )  ea
(2.24)
where D is the vapor pressure deficit (mbar); ea is the air vapor pressure (mbar); and
es(Ta) is the saturated air vapor pressure (mbar) at air temperature Ta (C).
2.2
Canopy photosynthesis
This section describes the amount of CO2 assimilated by the canopies.
Essentially, calculations involve determining the solar irradiance within the canopies,
calculating the CO2 assimilation rate by individual leaves, then scaling up to canopy
assimilation rate.
2.2.1 PAR components within canopies
There are four flux components within the canopies, and they are:
Q p, dr  (1  p)Qdr exp  kdr  L 

(2.25)
Q p, dr , dr  (1  p)Qdr exp   kdr  L
Q p, dr , 

(2.26)
Q p, dr  Q p, dr, dr 
(2.27)
2


(1  p)Qdf 1  exp   kdf L 


Q p, df 
 kdf L
(2.28)
where
Canopies flux components
Qp,dr is the PAR irradiance for unintercepted beam with scattered beam (mol photons
m-2 ground area s-1); Qp,dr,dr is the PAR irradiance for unintercepted beam without
scattering (mol photons m-2 ground area s-1); Qp,dr, is the PAR irradiance of the
scattered component only (mol photons m-2 leaf area s-1); and Q p, df is the mean
diffuse irradiance (mol photons m-2 leaf area s-1).
Note: Both Qp,dr, and Q p, df are based on a per unit leaf area (not per unit ground
area as for the other two components: Qp,dr and Qp,dr,dr).
PAR direct and diffuse irradiance
Qdr  I dr  0.5  4.55
(2.29)
Qdf  I df  0.5  4.55
(2.30)
where Qdr and Qdf are, respectively, the hourly direct and diffuse PAR irradiance, both
expressed in mol photons m-2 ground area s-1; and Idr and Idf are the hourly direct and
diffuse solar irradiance, respectively (W m-2 ground area).
Note: Total PAR irradiance is assumed 50% of total solar irradiance. It is then
converted from unit W m-2 ground area to mol photons m-2 ground area s-1 by
multiplying it with 4.55.
49
Canopy extinction coefficient for direct PAR
kdr 
0.5
0.5

sin  cos 
(2.31)
where kdr is the canopy extinction coefficient for direct PAR;  is the solar height (or
solar elevation) (radians); and  is the solar inclination ( = /2 - ) (radians).
Note: random distribution of leaves in the canopies is assumed.
Canopy clustering coefficient (correcting for non-uniform distribution of canopies
in the aerial space)
  01 2 
0  
(2.32)


L  
ln  b  (1   b ) exp   kdr

kdr L 
(1   b )  

1
1  L 3 L  Lmax
L  Lmax
0
b  
(2.33)
(2.34)
where  is the canopy clustering coefficient; 0 is the canopy clustering coefficient
when the sun is at zenith (highest point in the sky); L is the total leaf area index (m2
leaf area m-2 ground area); Lmax is the maximum leaf area index for full canopy cover
(taken as 4 m2 m-2);  is the solar inclination (radians); and kdr is the canopy extinction
coefficient for direct PAR.
Canopy extinction coefficient for diffuse PAR
kdf 
0.6936  13.2910 L2  2.3648 L4  0.1701L6
1  22.6522 L2  4.0257 L4  0.2674 L6
(2.35)
where kdf is the canopy extinction coefficient for diffuse PAR; and L is the total leaf
area index (LAI) (m-2 leaf area m-2 ground area).
Note: kdf has already been corrected for canopy clustering. Like that for kdr, the
assumption in kdf is also that leaves are randomly distributed in the canopies.
Canopy reflection and leaf absorption coefficients for PAR
Canopy reflection coefficient for PAR, p, is taken as 0.04, and the leaf scattering
coefficient for PAR, , is taken as 0.8.
2.2.2 PAR absorption by the leaves

Qsl   kdr Qdr  Q p, df  Q p, dr ,

Qsh   Q p, df  Q p, dr ,

(2.36)

(2.37)
where Qsl and Qsh are the PAR flux densities absorbed by the sunlit leaves and shaded
leaves, respectively (mol photons m-2 leaf area s-1); kdr is the canopy extinction
coefficient for direct PAR; Qdr is the hourly direct PAR irradiance (mol photons m-2
ground area s-1); Qp,dr, is the PAR irradiance of the scattered component only (mol
50
photons m-2 leaf area s-1); Q p, df is the mean diffuse irradiance (mol photons m-2
leaf area s-1); and  is the leaf scattering coefficient (where in this case, it is also the
leaf absorption coefficient) for PAR (0.8).
Note: PAR flux densities absorbed (Qsl and Qsh) are based on a per unit leaf area (not
per unit ground area).
2.2.3 Sunlit and shaded leaf area index
1  exp(kdr  L)
Lsl 
kdr 
(2.38)
Lsh  L  Lsl
(2.39)
2
where Lsl and Lsh are sunlit and shaded leaf area index, respectively (m leaf area m-2
ground area); L is the total leaf area index (m2 leaf area m-2 ground area); kdr is the
canopy extinction coefficient for direct PAR; and  is the canopy clustering
coefficient
2.2.4 Canopy temperature
Tf 
H c rac   H s  H c  raa
cp
 Tr
(2.40)
where Tf is the canopy (foliage) temperature (C); Tr is the air temperature at
reference level (C); Hs and Hc are the soil and crop sensible heat fluxes, respectively
(W m-2); raa is the aerodynamic resistance between the mean canopy flow and
reference level (s m-1); rca is the bulk boundary layer resistance (s m-1); and cp is the
volumetric heat capacity for air (1221.09 J m-3 K-1).
2.2.5 Temperature-dependent parameters
There are some parameters whose values are sensitive to the canopy/foliage
temperature Tf. Consequently, their values are adjusted according to
(T  25)/10
  (25) Q10 f
where (25) is the parameter value at 25 ºC (Kc(25), Ko(25), (25) or Vc,max(25)); Tf is the
canopy temperature (ºC); and Q10 is the relative change in the parameter  for every
10 ºC change. Values for the model parameters at 25 ºC and their respective Q10
values are given below.
Parameter values for the photosynthesis calculations. Parameters with subscript (25)
denote their values at 25 ºC.
(25)
Kc(25)
Ko(25)
(25)
Vc,max(25)
Description
Michaelis-Menten constant
for CO2
Michaelis-Menten constant
for O2
CO2 / O2 specificity factor
Rubisco capacity rate
Value
Unit
300 mol mol-1
Q10
2.1
300000 mol mol-1
1.2
2600 mol mol-1
200 mol m-2 s-1
51
0.57
2.4
Michaelis-Menten constant for CO2
(T f  25)/10
Kc  300  2.1
(2.41)
where Kc is the Michaelis-Menten constant for CO2 (mol mol-1); and Tf is the canopy
temperature (ºC)
Michaelis-Menten constant for O2
(T f  25)/10
Ko  300000  1.2
(2.42)
where Ko is the Michaelis-Menten constant for CO2 (mol mol-1); and Tf is the canopy
temperature (ºC)
CO2/O2 specificity factor
  2600  0.57
(T f  25)/10
(2.43)
where  is the CO2/O2 specificity factor; and Tf is the canopy temperature (ºC)
CO2 compensation point
O
*  a
2
(2.44)
where * is the CO2 compensation point (mol mol-1); Oa is the ambient O2
concentration in air (210000 mol mol-1); and  is CO2/O2 specificity factor.
Rubisco capacity rate
(T  25)/10
200  2.4 f
Vc,max 
1  exp 0.29 T f  40 




(2.45)
where Vc,max is the Rubisco capacity rate (mol m-2 leaf area s-1); and Tf is the canopy
temperature (ºC).
Note: Vc,max is corrected additionally for temperatures greater than 40 ºC after which
causes a rapid decline in the capacity rate due to Rubisco degradation.
2.2.6 Gross leaf CO2 assimilation

 sl / sh  MIN vc , vq, sl / sh , vs

(2.46)
where sl/sh is the gross leaf CO2 assimilation (mol CO2 m-2 leaf area s-1) for either
sunlit (subscript sl) or shaded (subscript sh) leaves; and MIN() is a function to select
the minimum of the three assimilation rates (vc, vq,sl/sh and vs).
Rubisco-limited assimilation (vc)
vc 
Vc,max  Ci   *
Kc 1  Oa Ko   Ci
(2.47)
where vc is the Rubisco-limited leaf CO2 assimilation rate (mol CO2 m-2 leaf area
s-1); Kc is the Michaelis-Menten constant for CO2 (mol mol-1); Ko is the Michaelis-
52
Menten constant for CO2 (mol mol-1); * is the CO2 compensation point (mol
mol-1); Oa is the ambient O2 concentration in air (210000 mol mol-1); Ci is the
intercellular CO2 concentration (245 mol mol-1); and Vc,max is the Rubisco capacity
rate (mol m-2 leaf area s-1).
Note: Vc,max, *, Kc and Ko must be corrected for canopy temperature.
Light-limited assimilation (vq)
vq, sl / sh  em Qsl / sh
Ci   *
Ci  2 *
(2.48)
where vq,sl/sh is the light-limited leaf CO2 assimilation rate (mol CO2 m-2 leaf area s-1)
for either sunlit (subscript sl) or shaded (subscript sh) leaves; * is the CO2
compensation point (mol mol-1); Ci is the intercellular CO2 concentration (245 mol
mol-1); Qsl/sh is the PAR absorbed by either sunlit (subscript sl) or shaded (subscript
sh) leaves (mol m-2 leaf area s-1); α is the leaf absorption of PAR (0.8); and em is the
intrinsic quantum efficiency or quantum yield (0.08).
Note: * must be corrected for canopy temperature.
Sink-limited assimilation (vs)
vs 
Vc,max
(2.49)
2
where vs is the sink-limited leaf CO2 assimilation rate (mol CO2 m-2 leaf area s-1);
and Vc,max is the Rubisco capacity rate (mol m-2 leaf area s-1).
Note: Vc,max must be corrected for canopy temperature.
2.2.7 Gross canopy CO2 assimilation
canopy   sl Lsl   sh Lsh
(2.50)
where canopy is the gross canopy CO2 assimilation (mol m-2 ground area s-1); sl and
sh are the gross leaf CO2 assimilation rates for sunlit and shaded leaves, respectively
(mol m-2 leaf area s-1); and Lsl and Lsh are the sunlit and shaded leaf area index,
respectively (m2 leaf area m-2 ground area).
2.2.8 Daily integration
Numerical method Gaussian integration is used to obtain the daily gross
canopy photosynthesis. Five points (N=5) over the diurnal period (from sunrise to
sunset) are selected, and for each selected hour, the gross canopy CO2 assimilation is
calculated.
Note: The daily gross canopy photosynthesis, canopy,d, must be converted from mol
CO2 m-2 ground area day-1 to g CH2O m-2 ground area day-1 before proceeding to
determine the plant respiration rates.
2.3
Energy Balance
The energy balance of the soil-plant-atmosphere system is described using the
electrical network analogy, where heat fluxes traverse a series of resistances to reach a
reference level above the canopies. Such a network is as shown in Fig. 2.1.
53
Fig. 2.1. Energy balance described as a network of resistances. Key: ET and ETc
and ETs are the latent heat fluxes from the system, crop and soil, respectively; H, Hc
and Hs are the sensible heat fluxes from the system, crop and soil, respectively; Rn is
the net radiation flux into the system; G is the heat conduction into the soil; Tr, Tf, To
and Ts are the temperatures for the reference height, crop, mean canopy flow and soil,
respectively; er, es and eo are the vapor pressure at the reference height, soil surface
and mean canopy flow, respectively; es(T) is saturated vapor pressure at temperature
T; raa is the aerodynamic resistance between the mean canopy flow and reference
height; rsa is the aerodynamic resistance between the soil and mean canopy flow; rca is
the bulk boundary layer resistance; rcs and rss are the canopy and soil surface
resistance, respectively.
2.3.1 Net radiation
Rn  1  p  It  RnL
(2.51)
 b  1  b  s DL 
RnL  RnL
(2.52)
  RLd  RLu
RnL
(2.53)
RLu   Ta  273.15
4
(2.54)
RLd  9.35  106  Ta  273.15
6
(2.55)
where Rn is the net radiation (W m-2); It is the hourly total solar irradiance (W m-2); p
is the surface albedo (typically, 0.15); RnL is the corrected net longwave radiation
(after adjusting for cloud cover) (W m-2); R'nL is the uncorrected net longwave
radiation (W m-2); s is the sunshine hours (hours); DL is the daylength (hours); b is an
54
empirical coefficient (between 0.1 and 0.3, taken as 0.2); RLd is the downward
longwave radiation (W m-2); RLu is the upward longwave radiation (W m-2); Ta is the
air temperature (C); and  is the Stefan-Boltzmann constant (5.67 x 10-8 W m-2 K-4).
2.3.2 Energy available to crop and soil
A  Ac  As  Rn  G


(2.56)
Ac  1   dr , Rn
(2.57)
As   dr , Rn  G
(2.58)

 dr ,  exp   kdr  L

(2.59)
where A is the net radiation available to the system (soil and crop); Ac is the fraction
of net radiation available to the crop (W m-2); As is the fraction of net radiation
available to the soil (W m-2); G is the soil heat flux (W m-2); L is the total leaf area
index (LAI); kdr is the canopy extinction coefficient for direct solar radiation;  is the
correction to the kdr due to canopy clustering, taken from Eq. (2.32)-(2.34); and α is
the scattering coefficient for total solar radiation (0.5).
2.3.3 Soil/ground heat flux


G  0.35  dr , Rn cos 
(2.60)
where G is the soil heat flux (W m-2); dr, is the penetration function from Eq. (2.59);
Rn is the net radiation (W m-2); and  is the solar inclination (radians).
2.3.4 Wind speed profile
This section describes properties related to how the wind speed varies with
height above and within the plant canopies.
2.3.4.1 Crop roughness length and zero plane displacement
d  0.64h
z0  0.26  h  d 
(2.61)
(2.62)
where d is the zero plane displacement height (m); z0 is the crop roughness length
(m); and h is the plant height (m).
2.3.4.2 Friction velocity
k  u ( h)
u* 
ln 1 0.26 
(2.63)
where u* is the friction velocity (m s-1); k is the von Karman constant (0.4); and u(h) is
the wind speed (m s-1) at the canopy top; that is, wind speed at plant height h.
Note: It is assumed that u(h) is equal to the daily wind speed measured by the weather
station.
55
2.3.4.3 Wind speed above and within the canopies
 u*  z  d 
 ln 

 k  z0 
u( z)  
 
z 
 u*  h  d 
 k ln  z  exp   1  h  


 0 

zh
zh
(2.64)
where u(z) is the wind speed (m s-1) at height z (m); h is the plant height (m); d is the
zero plane displacement height (m); z0 is the crop roughness length (m); k is the von
Karman constant (0.4); u* is the friction velocity (m s-1); and  is the wind speed
extinction coefficient (taken as 2).
2.3.4.4 Eddy diffusivity
 
z 
K ( z )  K (h)  exp  n 1   
  h 
(2.65)
K (h)  ku*h
(2.66)
where K(z) and K(h) are the eddy diffusivities at height z and at the canopy top,
respectively (m2 s-1); h is the plant height (m); k is the von Karman constant (0.4); u*
is the friction velocity (m s-1); and n is the eddy diffusivity extinction coefficient
(taken as 2).
2.3.5 Flux resistances
The heat fluxes have to traverse various types of resistances within the soilplant-atmosphere system. This section describes each of those resistance types.
2.3.5.1 Aerodynamic resistances
Soil surface to mean canopy flow
ras 
h  exp(n) 
 zs 0 
 z  d 
 exp  n 0
exp  n


n  K ( h) 
h 
h  


(2.67)
where ras is the resistance between the soil surface and the mean canopy flow (s m-1);
h is the plant height (m); k is the von Karman constant (0.4); u* is the friction velocity
(m s-1); n is the eddy diffusivity extinction coefficient (taken as 2); d is the zero plane
displacement height (m); z0 is the crop roughness length (m); and zs0 is the soil surface
roughness length (m)
Note: for flat, tilled land, zs0 can be taken as 0.004 m.
Mean canopy flow to reference level
raa 
 
z0  d   
1
1 
z d 
ln  r

exp  n 1 
  1

ku*  h  d  nku* 
h   
 
(2.68)
where raa is the resistance between the mean canopy flow and the reference level (s
m-1); h is the plant height (m); k is the von Karman constant (0.4); u* is the friction
velocity (m s-1); n is the eddy diffusivity extinction coefficient (taken as 2); d is the
56
zero plane displacement height (m); z0 is the crop roughness length (m); and zr is the
reference height (m).
2.3.5.2 Bulk boundary layer resistance
rac 

0.012 L 1  exp    2   u (h) w
(2.69)
where rac is the bulk boundary layer resistance (s m-1); L is the total leaf area index
(m2 leaf area m-2 ground area); w is the mean leaf width (m); u(h) is the wind speed at
the canopy top (i.e., at plant height h) (m s-1); and  is the wind speed extinction
coefficient (taken as 2).
2.3.5.3 Canopy resistance
a1  I PAR, t
rst 
a2 I PAR, t
rst L
rsc  
rst 0.5 Lcr
(2.70)
L  0.5Lcr
L  0.5Lcr
(2.71)
where rsc is the canopy resistance (s m-1); rst is the leaf stomatal resistance (s m-1); L
is the total leaf area index (m2 leaf area m-2 ground area); Lcr is the critical leaf area
index (m2 leaf area m-2 ground area, taken as 4); and IPAR,t is the total PAR irradiance
(W m-2); a1 and a2 are empirical coefficients, dependent on the crop type.
Note: Total PAR irradiance is assumed 50% of total solar irradiance: IPAR,t = 0.5It.
2.3.5.4 Soil surface resistance
l
rss, dry 
 p Dm, v
 1 v,1
rss  rss, dry exp   
  v, sat

(2.72)




(2.73)
where rss is the soil surface resistance (s m-1); Dm,v is the vapor diffusion coefficient
in air (24.7 x 10-6 m2 s-1);  is soil tortuosity (can be taken as 2); p is soil porosity; l is
the dry soil layer thickness (taken as the first soil layer thickness) (m); v,1 is the
volumetric soil water content of the first soil layer (m3 m-3); v,sat is the volumetric
saturated soil water content (m3 m-3); and  is the soil pore-size distribution index.
Note: The soil profile is divided into two or more layers (see section 2.4.1).
2.3.6 Total latent heat flux
 ET  Cc PM c  Cs PM s
(2.74)
57
raa  rac 


PM c 
   1  rsc  raa  rac  


A    c p D  ras Ac   raa  ras 


PM s 
   1  rss  raa  ras 



A   c p D  rac As
Cc  1  Rc Ra Rs  Rc  Ra 
1
Cs  1  Rs Ra Rc  Rs  Ra 
1
(2.75)
(2.76)
(2.77)
(2.78)
Ra       raa
(2.79)
Rc       rac   rsc
(2.80)
Rs       ras   rss
(2.81)
where ET is the total latent heat flux (W m-2); raa is the aerodynamic resistance
between the mean canopy flow and reference height (s m-1); rsa is the aerodynamic
resistance between the soil and mean canopy flow (s m-1); rca is the bulk boundary
layer resistance (s m-1); rcs and rss are the canopy and soil surface resistance,
respectively (s m-1); A, As and Ac are energy available to the system (total), soil and
crop, respectively (W m-2);  is the slope of the saturated vapor pressure curve (mbar
K-1);  is the psychometric constant (0.658 mbar K-1); D is the vapor pressure deficit
(mbar); and cp is the volumetric heat capacity for air (1221.09 J m-3 K-1).
Note: To convert latent heat fluxes from unit W m-2 (equivalent to J m-2 s-1) to depth
of water per second (mm s-1), simply divide it by , the latent heat of vaporization of
water (2454000 J kg-1). For example, latent heat flux 120 W m-2 is equivalent to
120 J m-2 s-1
2454000 J kg-1

1000 mm
 water
kg m-3
 4.9  105 mm s-1
where the density of water, water, is approximately 1000 kg m-3.
2.3.7 Air vapor pressure deficit at the mean canopy flow
ra
D0  D  a  A        ET 
c p
(2.82)
where D0 is the vapor pressure deficit at the mean canopy flow (mbar); ET is the
total latent heat flux (W m-2); raa is the aerodynamic resistance between the mean
canopy flow and reference height (s m-1); A is the total energy available to the system
(W m-2);  is the slope of the saturated vapor pressure curve (mbar K-1);  is the
psychometric constant (0.658 mbar K-1); D is the vapor pressure deficit (mbar); and
cp is the volumetric heat capacity for air (1221.09 J m-3 K-1).
58
2.3.8 Latent heat fluxes for crop and soil
 ETs 
 ETc 
As   c p D0 ras

   rss  ras

(2.83)
ras
Ac   c p D0 rac

   rsc  rac

(2.84)
rac
where ETs and ETc are the soil and crop latent heat fluxes, respectively (W m-2); As
and Ac are energy available to the soil and crop, respectively (W m-2); rsa is the
aerodynamic resistance between the soil and mean canopy flow (s m-1); rca is the bulk
boundary layer resistance (s m-1); rcs and rss are the canopy and soil surface resistance,
respectively (s m-1);  is the slope of the saturated vapor pressure curve (mbar K-1); 
is the psychometric constant (0.658 mbar K-1); D0 is the vapor pressure deficit at the
mean canopy flow (mbar); and cp is the volumetric heat capacity for air (1221.09 J
m-3 K-1).
2.3.9 Sensible heat fluxes for crop and soil
Hs 
Hc 


 As rss  ras   c p D0

ras   rss  ras



(2.85)
 Ac rsc  rac   c p D0

rac   rsc  rac

(2.86)
where Hs and Hc are the soil and crop sensible heat fluxes, respectively (W m-2); As
and Ac are energy available to the soil and crop, respectively (W m-2); rsa is the
aerodynamic resistance between the soil and mean canopy flow (s m-1); rca is the bulk
boundary layer resistance (s m-1); rcs and rss are the canopy and soil surface resistance,
respectively (s m-1);  is the slope of the saturated vapor pressure curve (mbar K-1); 
is the psychometric constant (0.658 mbar K-1); D0 is the vapor pressure deficit at the
mean canopy flow, respectively (mbar); and cp is the volumetric heat capacity for air
(1221.09 J m-3 K-1).
2.3.10 Daily integration
Numerical method Gaussian integration is used to obtain the daily latent and
sensible heat fluxes. Knowing the daily latent heat fluxes will enable us to determine
the daily amount of water transpired (by plants) and evaporated (by soil). Five points
(N=5) over 24 hours in a day are selected, and for each selected hour, the heat fluxes
are calculated.
2.4
Soil water
For accurate simulation of the water content in the soil, the soil profile has to
be divided into two or more layers, and for each layer, the soil water content is
simulated. Simulation of soil water profile is intensive and time-consuming because it
59
involves repetitious calculations. This is because hydraulic conductivity is sensitive to
the soil water content.
2.4.1 Partitioning of soil profile into several layers
Fig. 2.2 shows that the soil profile is divided from the soil surface into six (N
= 6) consecutive layers, where soil layer i (i = 1 to 6) has a thickness of si (m), and the
depth from the soil surface to the middle of layer i is zi (m). Water flux into soil layer
i is denoted as qi (m day-1).
Fig. 2.2. Water flow in a soil profile which is divided into six successive layers
Assumptions
1. Every soil layer has equal soil properties with each other. This means that all
soil layers have the same soil texture and soil water retention properties.
2. No water table effect, which means that there is no capillary rise of water from
the groundwater into the soil profile.
3. No irrigation is supplied and no surface runoff occurs.
60
4. The soil below the last soil layer is uniformly wet and it has the same water
content as the last soil layer. This means that water flows out of the last soil
layer (i.e., q7) due to gravitational pull alone (i.e., no matric suction gradient).
2.4.2 Soil layer depth and cumulative thickness
zi 
N
 0.5  si  si 1 
i 1
where s0  0
(2.87)
where zi is the depth of soil layer i from the soil surface (m), taken from the middle of
that soil layer to the soil surface; N is the number of soil layers (taken as 6); si and si-1
are the thickness of soil layer i and i-1, respectively (m).
Cumulative thickness
Si 
i
 sj
(2.88)
j 1
where Si is the cumulative thickness of soil layer i (m); that is, the sum of the
thickness of layer i and all its preceding layers.
2.4.3 Actual evaporation
Ea  ETs  RD, e
RD, e 
(2.89)
1


1  3.6073 v,1 v, sat 


-9.3172
(2.90)
where Ea is the actual evaporation (m day-1); ETs is the potential evaporation (m
day-1); RD,e is the reduction factor for evaporation (ranging from 0 to 1); and v,1 and
v,sat are the current soil water content in the first soil layer (i=1) and saturated soil
water content, respectively (m3 m-3).
Note: The soil water content from the first soil layer (i=1), v,1, is taken because all
evaporation occurs from this layer only (as this first layer is in contact with the
atmosphere).
2.4.4 Actual transpiration
Ta  ETc  RD,t
(2.91)
1

 v, root  v, wp
RD, t  
 v, cr  v, wp

0





v, root  v, cr
v, wp  v, root  v, cr
(2.92)
v, root  v, wp
v, cr  v, wp  0.5 v, sat  v, wp

(2.93)
where Ta is the actual transpiration (m day-1); ETc is the potential transpiration (m
day-1); RD,t is the reduction factor for transpiration (ranging from 0 to 1); v,root is the
61
total volumetric water content in the root zone (m3 m-3); v,wp is the volumetric water
content at wilting point (m3 m-3); and v,sat is the volumetric water content at
saturation (m3 m-3).
The algorithm to determine the total volumetric water content in the root zone,
v,root, is as follows:
v, root 
1
n
 i
(2.94)
d root i 1
in

v, i  si
i  
v, i   si   Si  droot   i  n

(2.95)
where i is the water content in soil layer i (m); droot is the rooting depth (m); si is the
thickness of soil layer i (m); and Si is the cumulative thickness of soil layer i (m). The
amount of water in the root zone is the summation of water content from the first soil
layer (i=1) until the n-th layer. The value of n is such that in the n-th soil layer, the
following relationship is true:
 Sn 1
Sn    droot Sn   1
(2.96)
2.4.5 Root extraction of water
The algorithm to determine the amount of water extracted by roots in each soil
layer is as follows:
Tai  Ta i  i 1 
(2.97)
 j  1.8c j  0.8c j 2
(2.98)

c j  MIN 1, S j droot

(2.99)
where Tai is the amount of water extracted by roots in soil layer i (m day-1); droot is the
rooting depth (m); Sj is the cumulative thickness of soil layer j (m); and MIN() is the
minimum of the enclosed values.
2.4.6 Hydraulic conductivity
The hydraulic conductivity in soil layer i (ki, m day-1) is read from the chart
depicting the relationship between hydraulic conductivity and volumetric water
content (Fig. 2.3). No hysterisis effect is assumed.
2.4.7 Matric suction head
The soil matric suction in soil layer i (Hmi, m) is read from the chart depicting
the relationship between soil suction and volumetric water content (Fig. 2.4). No
hysterisis effect is assumed.
Note: at soil saturation, suction is taken as 0 m.
62
1.2
1
0.8
hydraulic
conductivity 0.6
(m day-1)
0.4
0.2
0
0
0.2
0.4
0.6
0.8
water content (m3 m-3)
1
Fig. 2.3. Relationship between hydraulic conductivity and volumetric soil water
content
10000
1000
100
log-scale,
suction (m)
10
1
0.1
0
0.1
0.2
0.3
0.4
water content (m3 m-3)
0.5
Fig. 2.4. Relationship between soil suction and volumetric soil water content
2.4.8 Gravity head
H gi  zi
(2.100)
where Hgi is the gravity head in soil layer i (m); and zi is the depth of the middle of
soil layer i from the soil surface (m).
2.4.9 Hydraulic gradient
Hi  H mi  H gi
(2.101)
where in soil layer i, Hi is the hydraulic gradient (m), Hmi is matric suction head (m),
and Hgi is the gravity head (m).
63
2.4.10 Water content
Water flux
Darcy’s law is used to describe the water flow in the soil. Water flow is taken
to occur from the middle of layer i-1 to the middle of layer i. Water flux into soil layer
i is thus:
i 1
 R  Ea  Tai

  H i  H i 1 
qi  k
 Tai 1  i  N
  zi  zi 1 
k
i  N 1
 N
ki 
(2.102)
 ki 1si 1    ki si 
si 1  si
(2.103)
where subscripts i-1 and i denote soil layer i-1 and i, respectively; q is the water flux
(m day-1); R is the amount of daily rainfall (m day-1); Ea is the actual daily soil
evaporation (occurs only from the first soil layer) (m day-1); Ta is the daily extraction
of water by roots (actual plant transpiration) (m day-1); k is the weighted average of
the hydraulic conductivities of layer i and i-1 (m day-1); k and s are the soil layer’s
hydraulic conductivity and thickness, respectively (m); and H is the hydraulic gradient
(m).
Note: Water flux out of the last soil layer (i = N) is denoted by qN+1, and it is merely
equal to kN because of the assumption that the soil below the last layer is uniformly
wet and it has the same water content as the last soil layer. Consequently, water flux is
only due to gravity gradient (no matric suction gradient). In this case, qN+1 = kN.
Net flux
qˆi  qi  qi 1
(2.104)
where qˆi is the net flux in soil layer i (m day-1); and qi and qi+1 are the water flux into
soil layer i and i+1, respectively (m day-1).
Water content
i,t 1  i,t  qˆi
(2.105)
where i,t and i,t+1 are the water content in soil layer i (m) at time step t and t+1,
respectively; and qˆi is the net flux in soil layer i (m day-1).
2.4.11 Daily integration
Hydraulic conductivity is sensitive to soil water content especially at near
saturation. Consequently, it is vital that the water fluxes are calculated using very
small time steps because using too large a time step (e.g., 1 day) may cause
unrealistically large fluctuations or changes in the soil water content.
To overcome this problem, the single time interval is divided into f number of
subintervals, where the daily water content is simulated at successive subintervals.
64
Note: The larger the f, the more accurate the simulation results, but the slower the
simulation run. The value of f depends on the hydraulic conductivity value, k; a large
k often means a large f is required for accurate results. Typically, if the time step size
is 1 day (i.e., t = 1), f should be more than 500 (i.e., about 170 seconds).
2.5
Plant growth development
Food produced via photosynthesis is now used for plant maintenance
(supporting processes for continual plant survival) and growth (synthesis of new
cells).
2.5.1 Growth development stage and development rate
s,t 1  s,t  r ,t t
(2.106)
where s,t and s,t+1 are the growth development stage at time t and subsequent time
step t+1; t is the interval for each time step (days); r,t is the growth development
rate (days-1) at time t.
Note: The scale for the growth development stage ranges from 0 to 2, where
milestones along this scale are shown below.
Scale
0
1
2
Growth stage
Planting
Flowering (anthesis)
Maturity
Growth development rate, r, is read from a chart which depicts its linear relationship
with air temperature, as exemplified below (Fig. 2.5).
0.08
0.06
r 0.04
0.02
0
0
10
20
30
40
temperature (°C)
Fig. 2.5. Dependence of growth development rate on air temperature
2.5.2 Temperature sum
Daily temperature sum

0
Tts  

Tavg  Tb
Tavg  Tb
Tavg  Tb
(2.107)
65
where Tts is the daily temperature sum (ºC day); Tb is the base temperature for the
crop (ºC); and Tavg is the mean of the minimum and maximum air temperature (ºC).
Cumulative temperature sum
The summation of each day’s temperature sum gives the cumulative
temperature sum, and it is calculated by
Tts   Tts, t
(2.108)
t
where Tts is the cumulative temperature sum (ºC day); and Tts,t is the temperature
sum at day t (ºC day).
Note: An example showing the determination of the daily temperature sum Tts and
cumulative temperature sum Tts (base temperature Tb is 5 ºC) which depends on the
average air temperature Tavg.
Day
Tavg
Tts
Tts
1
4
0
0
2
5
0
0
3
8
3
3
4
10
5
8
5
12
7
15
6
10
5
20
7
8
3
23
8
5
0
23
9
8
3
26
2.5.3 Maintenance respiration
  2Ta  25 10  Wgreenleaves Wleaves
RM   RM


 
RM

i  plant parts
kM , iWi

 

(2.109)
 kM , greenleavesWgreenleaves  kM , stemWstem 
 kM , rootsWroots    kM , storageWstorage 
where RM is the maintenance respiration requirement (g CH2O m-2 ground area day-1);
kM,i is the maintenance respiration coefficient (g CH2O g-1 dry matter day-1) for plant
part i; and Wi is the weight (g dry matter m-2 ground area) for plant part i; and Ta is the
air temperature (C).
Maintenance respiration coefficient (at 25 C air temperature) for various plant parts
Plant part
green leaves, kM,greenleaves
stem, kM,stem
roots, kM,roots
storage organs, kM,storage
g CH2O g-1 dry matter day-1
0.030
0.015
0.015
0.010
Note: Maintenance respiration is corrected for air temperature and plant age.
66
2.5.4 Growth respiration

G
i  plant parts
Fi Gi


 Fgreenleaves Ggreenleaves   Fstem Gstem  
 Froots Groots    Fstorage Gstorage 
(2.110)
where G is the total glucose requirement (g CH2O g-1 dry matter); Gi is the glucose
requirement for each plant part i (i.e., green leaves, stem, roots and storage organs) (g
CH2O g-1 dry matter); and Fi is the fraction of dry matter of the individual plant parts.
General glucose requirement for the synthesis of various plant parts
g CH2O g-1 dry matter
1.463
1.513
1.444
1.415*
Plant part
green leaves, Ggreenleaves
stem, Gstem
roots, Groots
storage organs, Gstorage
* highly variable, dependent on crop type
2.5.5 Weight of plant parts (except leaves)
  canopy, d  RM
Wi, t 1  Wi, t  Fi 

G


 t

(2.111)
where Wi,t and Wi,t+1 are the weights (g m-2 ground area) of a given plant part i (stem,
roots or storage organs) at time step t and subsequent time step t+1, respectively; t is
the interval for each time step (days); canopy,d is the daily gross canopy CO2
assimilation (g CH2O m-2 ground area day-1); RM is the maintenance respiration
requirement (g CH2O m-2 ground area day-1); G is the total glucose requirement (g
CH2O g-1 dry matter); and Fi is the fraction of dry matter of the individual plant parts.
Note: The weight gain of leaves is calculated differently due to leaf death.
2.5.6 Leaf death

Dleaves  MAX Dage , Dshade

(2.112)
where Dleaves is the actual death rate of leaves (day-1); and MAX() is the maximum of
the enclosed values. As shown above, there are two possible reasons for leaf death: 1)
leaf age (Dage), and 2) self-shading of leaves (Dshade).
Death due to age

0

 
Dage   r
 2  s

 r
 0.1
 2   s   1.0
0.1   2   s   1.0
(2.113)
 2   s   0.1
67
where Dage is the leaf death rate due to leaf age (day-1); and r and s are the growth
development rate (day-1) and growth development stage (unitless), respectively.
Death due to self-shading
0


Dshade  
 L  Lcr  
MIN
0.03,
0.03



Lcr 


L  Lcr
L  Lcr
(2.114)
where Dshade is the leaf death rate due to leaf self-shading (day-1); L and Lcr are the
LAI and critical LAI, respectively (m2 m-2); and MIN[] is function to determine the
minimum of the enclosed values.
Note: Lcr is taken as 4 m2 m-2.
2.5.7 Weight of leaves
Dead leaves

Wdeadleaves, t 1  Wdeadleaves, t  Wgreenleaves, t Dleaves, t  t

(2.115)
where Wdeadleaves,t and Wdeadleaves,t+1 are the weights (g m-2 ground area) of dead leaves
at time step t and subsequent time step t+1, respectively; t is the interval for each
time step (days); and Dleaves,t is the death rate of leaves at time step t (day-1).
Green (alive) leaves
Wgreenleaves, t 1  Wgreenleaves, t 

  canopy, d  RM  
 Fgreenleaves 
  
G


  t
W

 greenleaves, t Dleaves, t

(2.116)
where Wgreenleaves,t and Wgreenleaves,t+1 are the weights (g m-2 ground area) of green
leaves at time step t and subsequent time step t+1, respectively; t is the interval for
each time step (days); Dleaves,t is the death rate of leaves at time step t (day-1); canopy,d
is the daily gross canopy CO2 assimilation (g CH2O m-2 ground area day-1); RM is the
maintenance respiration requirement (g CH2O m-2 ground area day-1); G is the total
glucose requirement (g CH2O g-1 dry matter); and Fgreenleaves is the fraction of dry
matter of green leaves.
2.5.8 Leaf area index
Lt  Wgreenleaves,t  SLAt
(2.117)
where Lt is the leaf area index (m2 m-2) at time step t; Wgreenleaves,t is the weight of
green leaves (g m-2 ground area) at time step t; and SLAt is the specific leaf area (m2
g-1, taken as 0.022 m2 g-1) at time step t.
2.5.9 Plant height
dh
ht 1  ht 
t
dt
(2.118)
68
b b h exp  b1Tts 
dh
 Tts  0 1 m
2
dt
1  b0 exp  b1Tts 
(2.119)
where ht and ht+1 are the plant heights (m) at time t and subsequent time step t+1; t is
the interval for each time step (days); Tts is the accumulated temperature sum (ºC
day); Tts is the temperature sum (ºC day); hm is the maximum possible height of the
plant (m); and b0 and b1 are the intercept (unitless) and slope (ºC-1 day-1) coefficients,
respectively.
2.5.10 Root elongation
v, t  v, wp or  s  1
d root , t

d root , t 1  
MIN  d m , d root , t  d g t  v, t  v, wp and  s  1






(2.120)
where droot,t and droot,t+1 are the rooting depth (m) at time step t and t+1, respectively;
dm is the maximum rooting depth (m); dg is a constant root elongation rate, denoting
rooting depth increase per day (m day-1, taken as 0.012 m day-1); t is the time step
interval (days); s is the growth development stage; v,t is the volumetric soil water
content at time t (m3 m-3); v,wp is volumetric soil water content at permenant wilting
point (m3 m-3); and MIN[] is the minimum function to determine the smallest of the
enclosed values.
Note: no root death is assumed.
2.5.11 Effects of water stress
Insufficient water limits growth according to the ratio between actual and
potential plant transpiration:
RD, t 
Ta
ETc
(2.121)
where RD,t is the reduction factor for growth (ranging from 0 for zero growth to 1 for
no growth suppression due to water stress); and Ta and ETc are the actual and potential
transpiration rates, respectively (m day-1).
The amount of assimilates potentially produced (i.e., in the absence of water
stress) is reduced in proportion to the level of water stress:

canopy
, d  canopy, d  RD,t
(2.122)
and plant height growth rate is reduced to
dh  dh

 RD, t
dt
dt
(2.123)
as well as root elongation rate to
d g  d g  RD,t
(2.124)
69
2.6
Summary
This section describes the flow or order of calculations to simulate the growth
of a general crop. It also list the symbols used throughout this report and the model
parameters required. The various components of the model are shown in Fig. 2.6.
Fig. 2.6. Overview of the gcg model components
70
2.6.1 Calculations flow
This section summarizes, using flowcharts (Fig. 2.7 to 2.15), the flow of
calculations as described in the earlier sections.
2.6.1.1 Main flow
Fig. 2.7. Main program flow for the generic crop growth model
71
2.6.1.2 Meteorology
Fig. 2.8. Program flow to determine the daily weather properties
72
Fig. 2.9. Program flow to determine the hourly weather properties
73
2.6.1.3 Photosynthesis
Fig. 2.10. Program flow to determine the daily gross canopy photosynthesis
74
Fig. 2.11. Program flow to determine the gross canopy assimilation rate
75
2.6.1.4 Energy balance
Fig. 2.12. Program flow to determine the daily evapotranspiration
76
Fig. 2.13. Program flow to determine the various heat fluxes
77
2.6.1.5 Soil water
Fig. 2.14. Program flow to determine the soil water content
78
2.6.1.6 Respiration
Fig. 2.15. Program flow to determine the maintenance and growth respiration rates
and the gains in the plant parts’ weights, leaf area index, plant height and rooting
depth
2.6.2 Model parameters
The following are parameters that must be measured or supplied to the model:
1. Meteorological properties:
a. Daily weather properties: sunshine hours (s), min. and max. air
temperature (Tmin and Tmax), relative humidity (RH), wind speed (u) and
rain (R).
b. Site latitude () and day of year (td).
c. Reference height (zr) at which meteorological properties were
measured.
79
2. Coefficients:
a. Angstrom coefficients, b0 and b1. These (b0, b1) parameters are
dependent on climate zones and can be taken generally as (0.18, 0.55)
for cold temperate zones, (0.25, 0.45) for dry tropical zones, and (0.29.
0.42) for humid tropical zones.
b. Surface albedo (p) which is typically 0.15 for most surfaces.
c. Empirical coefficient, b, for correcting net radiation for cloud cover. It
varies between 0.1 to 0.3. As an average, 0.2 can be taken.
d. Extinction coefficients for wind speed () and eddy diffusivity (n).
Both can be taken as 2.
e. Soil surface roughness length (zs0). For flat, tilled land, it can be taken
as 0.004 m.
f. Empirical coefficients a1 and a2 for the relationship between stomatal
resistance and PAR irradiance.
3. Plant properties:
a. Initial values for plant height (h), rooting depth (droot) and dry weights
(W) for plant parts such as stem, leaves, roots and storage organs.
b. Mean leaf width (w).
c. Root elongation rate (dg) and maximum rooting depth (dm).
d. Critical LAI (Lcr) and LAI for maximum canopy cover (Lmax). Both can
be 4 and 3 m2 m-2, respectively.
e. Specific leaf area (SLA).
f. Intercellular CO2 concentration (Ci) which can be taken as 245 mol
mol-1.
g. Intrinsic quantum efficiency or quantum yield (em) where for most
plants, it can be taken as 0.08 mol mol-1.
h. Relationship between growth development rate (r) and air
temperature.
i. Base temperature for crop growth (Tb).
j. Maintenance respiration coefficient (kM) and glucose requirement for
growth (G) for the various plant parts.
k. Plant height properties: maximum height (hm), growth coefficients (b0
and b1).
4. Soil properties
a. For each soil layer within the soil profile, the following must be
specified: its thickness (si) and initial volumetric soil water content
(v).
b. For the first soil layer, the following must be specified additionally:
pore-size distribution index (), porosity (p) and tortuosity (). Soil
tortuosity can be taken as 2.
80
c. All soil layers are assumed equal in the following specified properties:
i. Saturated and wilting point volumetric soil water content (v,sat
and v,wp)
ii. Relationship between hydraulic conductivity (k) and matric
suction head (Hm) with volumetric soil water content.
2.6.3 Main symbols
The main symbols used in this chapter are summarized below. Note that some
symbols (such as  and ) have more than one meaning, depending on their context.
Symbol



















*
0
canopy
canopy,d
cp
dr,
ET
ETc
Description
extinction coefficient for wind speed
fraction of PAR absorbed (0.8)
scattering coefficient (0.5 for total
solar radiation, or 0.8 for PAR)
solar elevation or height (from
horizontal)
solar declination
solar azimuth (from North)
psychometric constant (0.658)
latent heat of vaporization of water
(2454000)
pore-size distribution index
site latitude
pi constant (3.14159)
solar inclination (from vertical)
hour angle
CO2 / O2 specificity factor (2600)
soil tortuosity
Stefan-Boltzmann constant (5.67 x
10-8)
canopy clustering coefficient
slope of the saturated vapor pressure
curve
soil water content, expressed as water
depth
CO2 compensation point
canopy clustering coefficient when
the sun is at zenith
gross canopy assimilation rate of CO2
daily gross canopy assimilation rate
of CO2
volumetric heat capacity of air
(1221.09)
penetration function for direct solar
radiation (corrected for scattering)
total latent heat flux density
latent heat flux density of crop
81
Unit
radians
radians
radians
mbar K-1
J kg-1
radians
radians
radians
mol mol-1
W m-2 K-4
mbar K-1
m
mol mol-1
mol CO2 m-2 ground area s-1
g CH2O m-2 ground area day-1
J m-3 K-1
W m-2
W m-2
ETs
p
r
s
sh
sl
t
v
v,cr
v,root
v,sat
v,wp
A
Ac
As
b0, b1
b0, b1
Ci
d
dg
dm
droot
D
D0
Dage
DL
Dleaves
Dm,v
Dshade
ea
em
es(Ta)
Ea
ETc
ETs
Fi
G
G
latent heat flux density of soil
soil porosity
growth development rate
growth development stage
gross leaf assimilation rate of CO2 for
shaded leaves
gross leaf assimilation rate of CO2 for
sunlit leaves
size of each time step
volumetric water content
critical volumetric water content
volumetric water content in the root
zone
saturated volumetric water content
wilting point volumetric water content
total energy supply
fraction of energy available to the
plant
fraction of energy available to the soil
Angstrom coefficients
plant height growth coefficients
intercellular CO2 concentration in air
(245)
zero plane displacement
root elongation rate
maximum rooting depth
rooting depth
vapor pressure deficit
vapor pressure deficit at the mean
canopy flow
leaf death due to age
day length
leaf death
vapor diffusion coefficient in air (24.7
x 10-6)
leaf death due to self-shading
air vapor pressure
quantum efficiency (0.08)
saturated vapor pressure at air
temperature Ta
actual soil evaporation
potential transpiration
potential evaporation
dry matter fraction for plant part i
(green leaves, stem, roots or storage)
soil (ground) heat flux density
total glucose requirement for growth
82
W m-2
day-1
mol CO2 m-2 leaf area s-1
mol CO2 m-2 leaf area s-1
day
m3 m-3
m3 m-3
m3 m-3
m3 m-3
m3 m-3
W m-2
W m-2
W m-2
b0 (unitless), b1 (C day-1)
mol mol-1
m
m day-1
m
m
mbar
mbar
day-1
hour
day-1
m2 s-1
day-1
mbar
mol mol-1
mbar
m day-1
m day-1
m day-1
W m-2
g CH2O g-1 dry matter
Gi
h
hm
Hc
Hgi
Hi
Hmi
Hs
Ic
Idf
Idf,d
Idr
Idr,d
Iet
Iet,d
IPAR
It
It,d
k
ki
kdf
kdr
ki
kM,i
K(z)
Kc
Ko
l
L
Lcr
Lmax
Lsh
Lsl
n
Oa
p
qi
glucose requirement for growth for
plant part i (green leaves, stem, roots
or storage)
plant height
maximum plant height
sensible heat flux density of crop
gravity head
hydraulic gradient
matric suction head
sensible heat flux density of soil
solar constant
diffuse solar irradiance
daily diffuse solar irradiance
direct solar irradiance
daily direct solar irradiance
extraterrestrial solar irradiance
daily extraterrestrial solar irradiance
PAR irradiance (50% of It)
total solar irradiance
daily total solar irradiance
von Karman’s constant (0.4)
weighted average of hydraulic
conductivity for water flow into soil
layer i
canopy extinction coefficient for
diffuse solar radiation (corrected for
canopy clustering)
canopy extinction coefficient for
direct radiation
hydraulic conductivity for soil layer i
maintenance respiration coefficient
for plant part i (green leaves, stem,
roots or storage)
eddy diffusivity at height z
Michaelis-Menten constant for CO2
(300)
Michaelis-Menten constant for O2
(300000)
dry soil layer thickness
leaf area index (LAI)
critical LAI (equal to Lmax)
maximum LAI (4)
shaded leaf area index
sunlit leaf area index
extinction coefficient for eddy
diffusivity
ambient concentration of O2 in air
(210000)
surface albedo (reflection)
water flux into soil layer i
83
g CH2O g-1 dry matter
m
m
W m-2
m
m
m
W m-2
W m-2
W m-2
J m-2 day-1
W m-2
J m-2 day-1
W m-2
J m-2 day-1
W m-2
W m-2
J m-2 day-1
m day-1
-
m day-1
g CH2O g-1 dry matter day-1
m2 s-1
mol mol-1
mol mol-1
m
m2 leaf area m-2 ground area
m2 leaf area m-2 ground area
m2 leaf area m-2 ground area
m2 leaf area m-2 ground area
m2 leaf area m-2 ground area
mol mol-1
m day-1
qˆi
Q10
Qdf
Qdr
Q p, df
Qp,dr
Qp,dr,
Qp,dr,dr
Qsh
Qsl
ra a
rc a
rc s
rsa
rss
rss,dry
rst
R
R’M
R’nL
RD,e
RD,t
RH
RLd
RLu
RM
Rn
RnL
s
si
Si
SLA
td
th
tsr
net water flux into soil layer i
m day-1
relative change in a parameter for
every 10 ºC change
diffuse PAR component (above
canopies)
direct PAR component (above
canopies)
mean diffuse component of PAR
(within canopies)
unintercepted direct component of
PAR (with scattering component)
(within canopies)
PAR scattered component only
(within canopies)
unintercepted direct component of
PAR (without scattering component)
(within canopies)
total PAR absorbed by shaded leaves
total PAR absorbed by sunlit leaves
aerodynamic resistance between the
mean canopy flow and reference
height
bulk boundary layer resistance
canopy resistance
aerodynamic resistance between the
soil surface and mean canopy flow
soil surface resistance
soil surface resistance of dry soil
stomatal resistance
rainfall
total maintenance respiration rate
(uncorrected)
net longwave radiation
reduction factor for evaporation
reduction factor for transpiration
relative humidity
downward longwave radiation
upward longwave radiation
total maintenance respiration rate
(corrected for plant age and air
temperature)
net solar radiation (uncorrected)
net longwave radiation (corrected)
sunshine hours
thickness of soil layer i
cumulative thickness of soil layer i
specific leaf area
day of year
local solar time
time of sunrise
-
84
mol m-2 leaf area s-1
mol m-2 leaf area s-1
mol m-2 leaf area s-1
mol m-2 leaf area s-1
mol m-2 leaf area s-1
mol m-2 leaf area s-1
mol m-2 leaf area s-1
mol m-2 leaf area s-1
s m-1
s m-1
s m-1
s m-1
s m-1
s m-1
s m-1
m day-1
g CH2O m-2 ground area day-1
W m-2
%
W m-2
W m-2
g CH2O m-2 ground area day-1
W m-2
W m-2
hour
m
m
m2 g-1
day
hour
hour
tss
Tts
Ta
Ta
Ta,i
Tavg
Tb
Tf
Tmax
Tmin
Tset
Tts
u*
u(h),
u(z)
vc
vq
vs
Vc,max
w
Wi
z
z0
zi
zr
zs0
time of sunset
accumulated temperature sum
air temperature
actual plant transpiration
root extraction of water by roots in
soil layer i
average air temperature
base temperature for crop growth
temperature of canopy
maximum air temperature
minimum air temperature
air temperature at sunset
daily temperature sum
friction velocity
wind speed at plant height h or at
height z
Rubisco-limited rate of CO2
assimilation
light-limited rate of CO2 assimilation
sink-limited rate of CO2 assimilation
Rubisco capacity rate (200)
mean leaf width
dry plant weight for plant part i (green
leaves, dead leaves, stem, roots or
storage)
height
surface roughness length
depth from soil surface to the middle
of soil layer i
reference height
roughness length of soil surface
85
hour
ºC day
ºC
m day-1
m day-1
ºC
ºC
ºC
ºC
ºC
ºC
ºC day
m s-1
m s-1
mol m-2 leaf area s-1
mol m-2 leaf area s-1
mol m-2 leaf area s-1
mol m-2 leaf area s-1
m
g m-2
m
m
m
m
m
Chapter 3. Building the generic crop growth model in Excel
3.1
Operator precedence in Excel
This chapter shows how the various calculations are implemented in Excel to
build the gcg (generic crop growth) model. It is assumed that you have at least a basic
understanding of Excel’s operators, formulas and functions as well as how to navigate
the cells within Excel.
There are several mathematical operators in Excel, and Table 3.1 lists some of
these operators.
Table 3.1. Mathematical operators in Excel
Operation
Excel’s symbolic operator
Exponentiation
^
Multiplication
*
Division
/
Addition
+
Subtraction
-
Less than
<
Greater than
>
Less than or equal to
<=
Greater than or equal to
>=
These operators are self-explanatory except perhaps for exponentiation which is an
operation to raise a given constant (called base) to the power of another number. For
example, a^b means that a is raised to the power b so that 2^3 gives 2*2*2 or 8.
Excel follows the standard precedence of operators: exponents and roots first,
followed by multiplication and division, and finally addition and subtraction. Any
operations enclosed by brackets are always done first. Use of brackets or parentheses
always overrides the operator precedence.
In Excel, if we type the following formula
=2+3*4-9/3
it gives 11 because the order of calculations are 3*4, then 9/3. Their answers are 12
and 3, respectively. Their difference: 12-3 gives 9 which is finally added to 2 to give
the answer as 11. In another example,
=2+3^2*4-9/3
gives 35 because the exponent 3^2 is done first to give 9 which is then multiplied by 4
to give 36. 9/3 is next calculated to give 3. This value is subtracted from 36 to give 33
which is finally added to 2 to give the final answer as 35. We can place brackets to
86
override the precedence of operators and to specify the order of calculations
explicitly. For example,
=2+3^(2*4)-9/3
gives 6560 because 2*4 within the brackets is done first to give 8 so that 3^8 gives
6561. Finally 2+6561-3 gives 6560.
Unfortunately, Excel has a nonstandard order of precedence for unary
operators. Unary operators are simply positive (+) or negative (-) that precedes a
value. For instance, +2 and -2 mean positive and negative values of 2, respectively. In
Excel, if we have the following formula
=-2^4
it gives 16 because the calculations are done as (-2)^4. However, this does not
conform to the standard order of operator precedence. The calculations ought to be
-(2^4) to give -16. To circumvent this problem, we must write the formula as
=-(2^4)
which would give us the correct answer as -16. This is an issue that we have to
remember when using Excel to perform calculations involving unary and
exponentiation operators.
3.2
Functions and formulas in Excel
The gcg model uses only a few of Excel’s vast collection of functions. Table
3.2 lists the functions used and includes a brief description of each of them. Most of
these functions are self-explanatory. Consider an example of a data set in Fig. 3.1.
Fig. 3.1. Using VLOOKUP function to find values in the table
Based on the above tabulated data:
VLOOKUP(4, A2:B11,2,FALSE)
returns 45
VLOOKUP(6, A2:B11,2,FALSE)
returns 233
VLOOKUP(6, A2:B11,2,TRUE)
returns 233
VLOOKUP(1.5, A2:B11,2,TRUE)
returns 23
VLOOKUP(1.5, A2:B11,2,FALSE)
returns #N/A
87
The first case is VLOOKUP(4, A2:B11,2,FALSE) means that we wish to locate 4 in the
first column of a table, where this table is located in cells A2 to B11. Once 4 is found
in the first column, the function returns the corresponding value in the second column
(index_number is 2) of the table. The value in the second column corresponding to 4
is 45 (cell B5). So the function returns the value 45.
Table 3.2. List of Excel functions used to build the gcg model
Type
Function name
Description
Mathematics
EXP(x)
Returns ex, where e is about 2.71828.
SQRT(x)
Returns
x.
PI()
Returns the value of  (about 3.14159).
RADIANS(x)
DEGREES(x)
Converts x from unit degrees to radians
(RADIANS) or from radians to degrees
(DEGREES)
Trigonometry
SIN(x)
COS(x)
ACOS(x)
Sine, cosine ir arccosine of x
Statistical
MIN(x1, x2, …)
MAX(x1, x2, …)
Returns the minimum or maximum of the
enclosed x values.
Lookup
VLOOKUP(value,
table_array,
index_number,
range_lookup)
Searches for value in the leftmost column
of table_array and returns the value on
the same row based on the index_number.
The last argument range_lookup is set to
FALSE if an exact match is required, else
set to TRUE to return the next largest
value that is less than value if an exact
match cannot be found. The table_array
must be sorted in an ascending order.
Logical
IF(logical_test,
value_if_true,
value_if_false)
Returns value_if_true if the
logical_test is TRUE, else it returns
value_if_false if logical_test if
FALSE.
AND(logical_test1,
logical_test2, …)
Returns TRUE if all the given logical tests
(logical_test1, logical_test2, and so
on) are TRUE. Otherwise it returns
FALSE.
NOT(logical)
Returns the opposite of logical. If
logical is TRUE, this function returns
FALSE, else it returns TRUE if logical
is FALSE.
The last argument, range_lookup, is either TRUE or FALSE. If it is FALSE,
an exact match must be found. This is why the last case, VLOOKUP(1.5,
A2:B11,2,FALSE), returns the error message #N/A (for not available) because the
value 1.5 is not found in the first column of the table. However, VLOOKUP(1.5,
88
A2:B11,2,TRUE)
returns 23 because the next largest value that is less than 1.5 in the
first column is 1. So the corresponding value 23 in the second column is returned.
It is important to ensure that the tabulated data is sorted in an ascending order
for VLOOKUP to return the correct result.
The IF function is called a logical function because what it returns is based on
the given criteria. For instance, consider the following equation:
 x  10 x  10
y
 x / 10 x  10
which means we add 10 to x if x is smaller than 10, else we divide x by 10, and return
to result to y. In Excel, let cell A1 store the value of x, and cell B1 the value of y. So
in cell B1, we write the formula:
=IF(A1<10,A1+10,A1/10)
where A1+10 is performed if A1 is less than 10, but if A1 is greater than or equal to
10, A1/10 is performed instead. In other words, if A1<10 is TRUE, A1+10 is returned,
but if A1<10 is FALSE, A1/10 is returned instead.
It is possible to nest the IF functions. For example, consider an expanded
equation:
x0
0

y   x  10 0  x  10
 x / 10 x  10

where y can now have three possible values, depending on the value of x. In cell B1,
we may write:
=IF(A1<=0,0,IF(A1>=10, A1/10, A1+10))
If A1<=0 is TRUE, 0 is returned, but if A1<=0 is FALSE, another IF function,
IF(A1>=10, A1/10, A1+10), is evaluated. In IF(A1>=10, A1/10, A1+10), if
A1>=10 is TRUE, A1/10 is returned, else it is A1+10 (that is, A1>=10 is FALSE).
The AND function returns either TRUE or FALSE only. If all its logical
arguments are TRUE, the AND function returns TRUE, else it returns FALSE if one or
more of its logical arguments are FALSE. Consider the following example in Fig. 3.2
to determine if x1 is greater than 0 and x2 less than 0.
Fig. 3.2. Illustrating the use of the AND function
Based on the above example, only the second case, where x1 is 1 and x2 is -2, that the
AND function returns TRUE because both conditions x1>0 and x2<0 are TRUE.
The AND function is particularly useful when it is used in conjunction with the
IF function. For instance, instead of writing the formula like this, as we did before,
89
=IF(A1<=0,0,IF(A1>=10, A1/10, A1+10))
we can write it alternatively as
=IF(A1<=0,0,IF(AND(A1>0,A1<10),A1+10,A1/10))
3.3
Basic setup: workbook and worksheets
A single workbook is required to build the gcg model. This workbook will
contain several worksheets, where each worksheet either stores tabulated data or
implements individual components of the model.

Create a new Excel workbook and save it as “gcg”. Unless you specify explicitly
the file extension, all workbook filenames in Excel 2003 will be saved with an
extension “.xls”, whereas in Excel 2007, the appended extension is “.xlsx”.

Depending on how Excel is setup in your computer system, your workbook will
have either one or more blank worksheets, and they are named sequentially as
Sheet1, Sheet2, Sheet3, and so on. If you have less than three worksheets in the
workbook, insert worksheets until you have at least three worksheets. We will
require more worksheets later as we gradually build up the crop growth model, but
as for now, only three worksheets are required.
3.3.1 Weather worksheet
The gcg model requires daily weather properties only. Hourly weather
properties will be simulated using the daily weather properties. Looking up a value
from the weather table is more efficient and convenient if all the site’s weather data
are stored in a single worksheet. This includes storing all the site’s weather data for
one or more years. For example, the weather data from Reading, UK for the years
2000 to 2004 should all be stored in a single worksheet. Avoid breaking up the
weather data set, for instance, into individual years and storing each year’s data in a
separate worksheet.
In this book, the sample weather data from Serdang, Malaysia (from 19852004) will be used as an example. This weather data can be downloaded from
www.agri.upm.edu.my/~chris/weather. The Serdang weather file obtained from the
website is formatted such that the first three columns are for year, month and day. The
next six columns are for daily weather properties: sunshine hours, maximum air
temperature, minimum air temperature, relative humidity, wind speed and rainfall.

Start up any text editor program (such as Windows Notepad) to open the Serdang
weather text file which you downloaded from the website. Select and copy all the
text (Fig. 3.3). Close the text editor program, if desired.

In the gcg workbook, rename worksheet Sheet1 to Serdang. This worksheet will
store the weather data for Serdang for all years 1985-2004.

Paste the clipboard contents (containing the Serdang weather) in cell A2. You will
see that the weather data are spread over columns A to I (with a blank first row).
90
Fig. 3.3. Open the Serdang weather text file using a text editor. Select all the text and
copy into the clipboard to be later imported into Excel.

Right click on column D, and choose Insert from the popup menu (Fig. 3.4). This
will insert a new column which will appear to the left of column D. After this
operation, the weather data will now be spread over columns A to J, with a new
blank column D.
Fig. 3.4. Insert a column for date at column D in the Serdang worksheet

The empty first row in the Serdang worksheet serves as a table header to describe
the individual weather properties. In cell A1, type year, followed by month, day
and date in the subsequent cells B1, C1, D1, respectively. The first four columns
denote the date of weather. As stated earlier, the daily weather consist of six
measured properties: sunshine hours (sunhr), maximum air temperature (tmax),
minimum air temperature (tmin), relative humidity (rh), wind speed (wind) and
rainfall (rain). Consequently, type the following table headers sunhr, tmax, tmin,
91
rh, wind
and rain in cells E1, F1, G1, H1, I1 and J1, respectively, to describe
these six weather properties (Fig. 3.5).
Fig. 3.5. The weather table formatted as required in the Serdang worksheet. Also
notice the other three worksheets (Control, Output and Meteorology).

At this moment, all columns from A to J are filled with data except the blank
column D (which we inserted previously). The purpose for having this column is
to specify the date of the weather. However, different countries specify dates in
differing formats. For instance, a date written as “02/12/08” may mean 2
December 2008, 12 February 2008 or even 12 August 2002. To avoid this
confusion, especially when the spreadsheet model is being executed in different
computer systems, Excel’s DATE function will be used to specify the date of the
weather correctly according to the computer’s regional settings. The
DATE(year,month,day) function has three arguments which correspond to the
year, month and day, respectively. Consequently, in cell D2, type the formula
=DATE(A2,B2,C2)
which will give the date (recall that columns A, B and C store the year, month and
day, respectively). Do the same for all other cells in column D. To avoid
repetitious and tedious copying-and-pasting, you can select cell D2, and double
click the fill handle (a small square at the bottom right corner of the cell). When
the mouse pointer moves over the fill handle, the mouse pointer changes into a
small cross (Fig. 3.6). Double clicking on the cell’s fill handle copies and pastes
the cell’s formula down the column.
92
Fig. 3.6. The fill handle is at bottom right hand corner of the cell. Double clicking it
will paste the cell’s contents down the column.
The gcg model requires that the weather data be sorted ascendingly according
to the date; that is, arranged from the earliest to the latest year, and within each year,
the data are sorted from 1 January to 31 December. The weather data must also be
complete and should not have “gaps” or missing data for certain days. If this occurs,
you have to interpolate between measured days. The formula for linear interpolation
is simply
x2  x1   x3  x1   t2  t1  t3  t1 
where t is the date (e.g., day of year) and x is the weather property (e.g., sunshine
hours). The missing data set (t2, x2) lies between two known sets, (t1, x1) and (t3, x3),
and the determination of the missing value x2 at t2 is by linear interpolation between
the two knowns x1 and x3 measured at t1 and t3, respectively.
3.3.2 Control worksheet
The Control worksheet is where we specify the model’s step number, step size
and the criteria to stop the iterations. In this worksheet, we will also have a list of
operations to run. BuildIt requires at least three names to be defined: _stepsize,
_step and _criteria. These names dictate to BuildIt the current step number
(_step), the size of each iteration step (_stepsize) and when to stop the iteration
runs (_criteria). Begin by renaming worksheet Sheet2 to Control.
Enter the following in the Control worksheet:
Row
1
2
Col. B
Cell name
10
B2:
maxsteps
3
Col. A
CONTROL
Number of
iterations
Step size
1
4
5
Current step
Criteria
0
=(_step/_stepsize) <
maxsteps
B3:
_stepsize
B4: _step
6
7
8
9
10
Starting date
Current date
Day of year
Current hour
=DATE(2000,1,1)
=B7+_step
=B8-DATE(YEAR(B8),1,0)
12
Eq. ref.
B5:
_criteria
B8: date
B9: doy
B10: th
The values in cells B2, B3 and B4 and their respective defined names mean
that maxsteps = 10, _stepsize = 1 and _step = 0, respectively. At the moment, their
values are inconsequential and will be changed as required by a simulation task.
Cell B5 specifies the criteria to BuildIt when to stop the model iteration runs.
As long as (_step/_stepsize) < maxsteps (cell B5 is TRUE), iteration continues
until (_step/_stepsize)  maxsteps (cell B5 is FALSE).
93
Notice that the simulation start date in cell B7 is specified as
=DATE(2000,1,1)
which means that the simulation start date is 1 January 2000. You should enter the
date using the DATE function, rather than entering “01/01/00” in cell B7. Recall that
this format (e.g., 01/01/00) may be interpreted differently on different computer
systems.
In cell B8, the formula is
=B7+_step
which means that it adds the simulation start date (in cell B7) to the model’s step
number (_step). BuildIt increments _step automatically, so as the iteration
continues, the current simulation date will be updated accordingly. For instance, take
the simulation start date as 1 Jan. 2000 and the _stepsize as 1. When the simulation
begins, _step will be set to 0, so the current simulation date is 1 Jan. 2000. In the
second iteration run, _step will be incremented by _stepsize; that is, _step =
_step + _stepsize to give _step = 1. This means the current simulation date is now
2 Jan. 2000. In the third iteration step, the _step is 2 to give the current simulation
date as 3 Jan. 2000, and so on.
In cell B9, the formula to determine the day of year for the current date is
=B8-DATE(YEAR(B8),1,0)
which gives the number of days from the first day of the same year (1 January).
However, since the day of year starts as 1 (not 0), the subtraction of dates must be
specified as B8-DATE(YEAR(B8),1,0) and not as B8-DATE(YEAR(B8),1,1). It might
seem odd that the day argument in DATE is given as 0 (e.g., there is no 0 January), but
doing so will ensure that the day of year for 1 January is 1 (not 0). We also use the
YEAR function to extract the year for the current simulation date (in cell B8), so that
the day of year will always be within the range of 1 and 365 (or 366 for leap years). If
we had typed the following in cell B9:
=B8-B7+1
works so long as the years for the current date (cell B8) and starting date (cell B7) are
not different. If the starting and current date are 1 January 2000 and 12 January 2001,
respectively, the calculated day of year would have been erroneously 378 (note: year
2000 is a leap year).
Note that because we are dealing with dates in cell B9, the result from the date
subtraction may be displayed as a date format (e.g., displayed as 1-Jan-00) in cell B9.
This can be confusing, so we have to change the display format to Number or
General. Right click on cell B9, and choose Format Cells from the popup menu.
Choose the Number tab, and ensure the display format is set either to General or
Number (Fig. 3.7). If the latter format is chosen, you can set the number of decimal
places to 0 since the day of year is an integer (i.e., no decimal places).
Lastly, cell B10 is the current local solar hour for the simulation.
94
Fig. 3.7. Format Cells dialog to format the display of cells
3.3.3 Output worksheet
We often want to store the model output results so that we can later analyze
and interpret the simulation results, and even plot charts from the results. We need to
tell BuildIt: 1) which data should be stored, and 2) where to store the results. For this
purpose, BuildIt requires two names to be defined: _read and _write. When a cell
name is given the name _read, the entire row for that cell is used by BuildIt to
identify cell contents that should be stored at every iteration run. Complementing
_read, the name _write is to tell BuildIt where the results should be displayed. When
a cell is given the name _write, this cell serves as a starting point to store the output
results. It is convenient if all model output results be stored in a separate worksheet,
called as the Output worksheet.

Rename the worksheet Sheet3 to Output.

Define the names _read to cell A2 and _write to cell A6.

In cell A5, type Model output:
We will leave the Output worksheet as it is now, and return to it later when we
have specific simulation tasks to run and their results to save.
3.4
Meteorology worksheet
The meteorology model component will be implemented in a separate
worksheet named Meteorology. Insert a new blank worksheet and rename it to
95
Meteorology. We will begin by specifying the model parameters (input) related to
meteorology.
Enter the following in the Meteorology worksheet:
Row
1
2
3
4
5
Col. A
INPUT
lat. (deg)
Angstrom, b0
Angstrom, b1
Weather file
Col. B
Cell name
3
0.29
0.42
Serdang
Eq. ref.
(2.7)
(2.7)
Site latitude (cell B2) is specified in degrees for Serdang, Malaysia. The
Angstrom coefficients, b0 and b1, for determining the daily total solar irradiance are
specified in cells B3 and B4, respectively, and their values are for a humid tropical
climate zone. Surface albedo is 0.15 and is specified in cell B5.
Lastly, cell B5 is the name of the worksheet that holds the weather data to be
used in simulation. Since we are using the Serdang weather which is stored in the
Serdang worksheet, we enter its worksheet’s name (Serdang) in cell B5. This is a
convenient feature that allows us to specify quickly which weather data set to use in
simulation. For example, we may have another weather data set for a different site
such as for Semenyih, Malaysia which is stored in a separate worksheet named as
Semenyih. Consequently, to use the Semenyih weather data set, we can enter
Semenyih in cell B5.
Excel functions require angles to be specified in unit radians, so we need to
convert the site latitude from degrees to radians. Conversion is simple by using the
function RADIANS (cell E2):
Row
2
3
Col. D
lat. (rad)
Weather ref.
Col. E
=RADIANS(B2)
=$B$5&"!D:J"
Cell name
Eq. ref.
E2: lat
Recall that cell B5 contains the name of the worksheet where the weather data
are stored. In our example, the weather is stored in the Serdang worksheet, so cell B5
contains the text “Serdang”. In cell E3, the formula
=B5&"!D:J"
joins (concatenates) the two texts, “Serdang” (cell B5) and “!D:J”, to produce a single
text “Serdang!D:J” which is a cell range reference to the weather table, and will be
used to lookup the weather table (see next).
3.4.1 Daily weather lookup
To look up values from the weather table, the function VLOOKUP is used. Recall
that this function has the following specification:
VLOOKUP(value,table_array,index_number,range_lookup)
which searches for value in the leftmost column of table_array and returns the
value on the same row based on the index_number. The last argument range_lookup
is set to FALSE if an exact match is required, else set to TRUE to return the next
largest value that is less than value if an exact match cannot be found.
The following entries in the Meteorology worksheet illustrate how this
function is used to lookup the data from the weather table:
96
Row
5
6
Col. D
DAILY WEATHER
sunshine hours
7
max. temp.
8
min. temp.
9
10
relative
humidity
wind speed
11
rain
12
mean temp.
Col. E
Cell name
=VLOOKUP(date,INDIRECT(E3)
,2, FALSE)
=VLOOKUP(date,INDIRECT(E3)
,3, FALSE)
=VLOOKUP(date,INDIRECT(E3)
,4, FALSE)
=VLOOKUP(date,INDIRECT(E3)
,5, FALSE)
=VLOOKUP(date,INDIRECT(E3)
,6, FALSE)
=VLOOKUP(date,INDIRECT(E3)
,7, FALSE)
=(tmax+tmin)/2
E6: sunhr
Eq. ref.
E7: tmax
E8: tmin
E9: rh
E10:
daywind
E11: rain
E12: tmean
For instance, to lookup the sunshine hours for 12 February 2000 for Serdang,
we may write
VLOOKUP(DATE(2000,2,12),Serdang!D:J,2,FALSE)
where the first argument is the value that we wish to find (i.e., 12 February 2000)
from the table in the Serdang worksheet in columns D to J (i.e., Serdang!D:J). If we
look in the Serdang worksheet (Fig. 3.5), we see that the entire Serdang weather table
(from 1985 to 2004) spreads from cell A2 to J7306. However, the first three columns
(A, B and C), storing the year, month and day, respectively, will not be used in table
lookup. Instead we will use the fourth column (D), storing the weather date, to lookup
a table. Consequently, our table for weather lookup actually begins from column D to
J. And since the entire Serdang worksheet is dedicated to storing the weather table, we
can eliminate the row numbers and write Serdang!D:J instead of
Serdang!D2:J7306.
In our example, once the date 12 February 2000 is located from the table, the
value from the second column (as specified by the third argument, index_number) of
the table is returned. Looking in the Serdang worksheet again (Fig. 3.5), we see that
the sunshine hours data are stored in column E which is the second column after
column D. The last argument is set to FALSE because an exact match (not an
approximate match) must be found. As another example, to lookup the maximum air
temperature (located in column F in the Serdang worksheet), we may type
VLOOKUP(DATE(2000,2,12),Serdang!D:J,3,FALSE)
Recall that in cell E3, the formula written was
=B5&"!D:J"
which concatenates the name of the weather worksheet (stored in cell B5) and the text
“!D:J”. So in our case, the resultant text in cell E3 is “Serdang!D:J”. But if we write
like this
VLOOKUP(DATE(2000,2,12),E3,2,FALSE)
it fails to lookup the sunshine hours from the Serdang weather table because the
second argument states that cell E3 is where the lookup table is located. Cell E3
contains the reference where the weather table is located and is not the lookup table
itself. To circumvent this problem, we use the INDIRECT function. This function
97
returns the reference to a cell based on its text representation, so writing the formula
like this
VLOOKUP(DATE(2000,2,12),INDIRECT(E3),2,FALSE)
is as if we had written like this
VLOOKUP(DATE(2000,2,12),Serdang!D:J,2,FALSE)
Consequently, for the current simulation date, daily weather properties will be
looked up from the weather table via the function
VLOOKUP(date,INDIRECT(E3),index_number,FALSE)
where index_number is 2 to 7, depending on the weather property, and date is the
name referring to cell B8 in the Control worksheet which holds the current simulation
date. Based on this date, the weather table (in the Serdang worksheet) will be looked
up for the daily weather properties.
Lastly, in cell E12, the average of the minimum and maximum air temperature
is calculated.
3.4.2 Solar position
Solar position is defined by its angle from vertical (solar inclination) and from
North in a clockwise direction (solar azimuth). Enter the following in the
Meteorology worksheet:
Row
14
15
Col. D
SOLAR POSITION
declination
16
17
18
19
20
hour angle
a
b
inc
suninc (rad)
21
22
suninc (deg)
solar height
23
c
24
25
26
azi
sunazi (rad)
sunazi (deg)
Col. E
Cell name
Eq. ref.
=-0.4093 *
COS(2*PI()*(doy+10)/365)
=(PI()/12)*(th-12)
=SIN(decl)*SIN(lat)
=COS(decl)*COS(lat)
=ACOS(E17+E18*COS(ha))
=IF(E19<PI()*0.5,E19,PI()*
0.5)
=DEGREES(suninc)
=PI()*0.5-suninc
E15: decl
(2.1)
E16: ha
(2.10)
E20: suninc
(2.11)
(2.11)
=(SIN(lat)*SIN(solarhgt)SIN(decl)) /
(COS(lat)*COS(solarhgt))
=IF(E23<1,ACOS(E23),0)
=PI()+IF(th<12,-E24,E24)
=DEGREES(sunazi)
E22:
solarhgt
(2.11)
(2.11)
(2.12)
E25: sunazi
(2.12)
(2.12)
(2.12)
Note that a, b and c are intermediary variables. Cell E19 contains the
calculated solar inclination (angle from vertical). However, if th is before sunrise or
after sunset, solar inclination will exceed /2 radians (or 90) because the sun is
below the horizon. In agriculture, calculating the solar position for hours before
sunrise or after sunset is of little interest. Consequently, the IF function in cell E20
ensures that the solar inclination is capped at /2 if its calculated value exceeds /2.
Another issue to deal with is the calculation of solar azimuth in cell E23. The
intermediary variable, c, can exceed 1 due to rounding errors by the computer. If the
arccosine of c is taken in this case, it will lead to an undefined value. To circumvent
98
this problem, the formula in cell E24 uses the IF function to check if c is less than 1
before returning the arccosine of c. Otherwise, the function returns 0 radian.
Cells E21 and E26 contain the angles converted to unit degrees for our
reference.
3.4.3 Sunrise, sunset and daylength
The following determines the local solar hour for sunrise and sunset, and the
difference between them gives the day length:
Row
28
29
Col. D
SUNRISE/SET
sunset
30
31
sunrise
daylength
Col. E
Cell name
Eq. ref.
=12+(12/PI())*ACOS(-E17 /
E18)
=24-E29
=tss-tsr
E29: tss
(2.3)
E30: tsr
(2.2)
(2.4)
E31: dl
Cells E17 and E18 refer to the intermediary variables, a and b, respectively, as
previously defined in section 3.4.2.
3.4.4 Air temperature
The current air temperature is calculated as follows:
Row
33
34
Col. D
AIR TEMP.
sunset temp.
35
Ta1
36
Ta2
37
Ta3
38
air temp.(°C)
39
air temp.(K)
Col. E
Cell name
Eq. ref.
=tmin+(tmax-tmin) *
SIN((PI()*(tss-tsr-1.5)) /
(tss-tsr))
=tset+((tmin-tset)*(24+thtss))/((tsr+1.5)+(24-tss))
=tmin+(tmax-tmin) *
SIN((PI()*(th-tsr-1.5))
/(tss-tsr))
=tset+((tmin-tset)*(thtss))/((tsr+1.5)+(24-tss))
=IF(th<(tsr+1.5),E35,IF(th
>tss,E37,E36))
=ta+273.15
E34: tset
(2.20)
(2.19)
(2.19)
(2.19)
E38: ta
(2.19)
E39: tak
(2.19)
Cells E35, E36 and E37 show that there are three possible values to air
temperature (i.e., Ta1, Ta2 and Ta3). Cell E38 chooses the correct value depending on
the current local solar hour (th) using the nested IF functions. Finally, cell E39
converts the air temperature from unit degrees Celsius to Kelvin.
3.4.5 Air vapor pressure and wind speed
Below determines the current air vapor pressure and saturated air vapor
pressure. Also recall that the wind speed is assumed constant throughout the whole
day.
Row
41
42
43
44
Col. D
AIR VAP. PRES.
sat. vap.
pressure
vap. pressure
Col. E
Cell name
Eq. ref.
=6.1078 *
EXP(17.269*ta/(ta+237.3))
=(rh/100)*E42
E42: es
(2.21)
E43: ea
(2.23)
99
45
46
WIND SPEED
wind speed
E46: wind
=daywind
section
2.1.2.6
3.4.6 Solar radiation
3.4.6.1 Daily total solar radiation
The following determines the daily total solar radiation and the partitioning
into its diffuse and direct components:
Row
1
Col. G
SOLAR
RADIATION
eccentricity
factor
solar constant
sinb
Col. H
Cell name
5
6
7
8
9
10
11
12
13
14
Daily ET
Daily total
=1+0.033*COS(2*PI()*(doy10)/365)
=1370*H2
=(24/PI())*(E17*ACOS(E17/E18)+E18*SQRT(1(E17/E18)^2))
=3600*H3*H4
=H5*(B3+B4*sunhr/dl)
total:et
f1
f2
f3
f4
=Itd/H5
=Itd
=Itd*(1-2.3*(H8-0.07)^2)
=Itd*(1.33-1.46*H8)
=Itd*0.23
Daily diffuse
15
Daily direct
=IF(H8<0.07,H9,IF(AND(H8>=
0.07,H8<0.35),H10,
IF(AND(H8>=0.35,H8<0.75),H
11,H12)))
=Itd-H14
2
3
4
Eq. ref.
(2.5)
(2.5)
(2.6)
H6: Itd
(2.6)
(2.7)
(2.8)
(2.8)
(2.8)
(2.8)
(2.8)
H14: Idfd
(2.8)
H15: Idrd
(2.9)
Note that cells E17 and E18 refer to the intermediary variables, a and b,
respectively, as defined previously in section 3.4.2. Cells H9 to H12 show that there
are four possible values for the fraction of diffuse solar component (i.e., f1, f2, f3
and f4). Cell H14 chooses the correct value depending on the ratio of total solar
irradiance to extraterrestrial solar irradiance using nested IF functions.
3.4.6.2 Hourly total solar radiation
The following determines the hourly solar radiation and the partitioning into
its diffuse and direct components:
Row
17
18
19
Col. G
ET
p1
p2
20
21
22
23
phi
phiA
phiB
total
Col. H
=H3*SIN(solarhgt)
=PI()*Itd/86400
=E17*ACOS(-E17/E18) +
E18*SQRT(1-(E17/E18)^2)
=H18/H19
=-E18*H20
=E17*H20
=H21*COS(2*PI()*th/24) +
H22
100
Cell name
Eq. ref.
(2.13)
(2.14)
(2.14)
(2.14)
(2.14)
(2.14)
(2.14)
24
25
26
27
Hourly total
=IF(H23<0,0,H23)
total:et
R
28
29
30
31
32
33
34
K
f1
f2
f3
f4
=It/H17
=0.847-1.61*SIN(solarhgt)
+ 1.04*(SIN(solarhgt)^2)
=(1.47-H27)/1.66
=It
=It*(1-6.4*(H26-0.22)^2)
=It*(1.47-1.66*H26)
=It*H27
35
Hourly direct
Hourly diffuse
=IF(It<=0,0,IF(H26<=0.22,H
29,IF(AND(H26>0.22,H26<=0.
35),H30,IF(AND(H26>0.35,H2
6<=H28),H31,H32))))
=It-H34
H24: It
(2.14)
(2.15)
(2.16)
(2.17)
(2.15)
(2.15)
(2.15)
(2.15)
H34: Idf
(2.15)
H35: Idr
(2.18)
Note that cells E17 and E18 refer to the intermediary variables, a and b,
respectively, as defined previously in section 3.4.2. Cell H23 calculates the total solar
irradiance for the current local solar hour. However, the calculated value can be
negative for hours before sunrise or after sunset. At these hours, there is no solar
irradiance (that is, it is zero). Consequently, cell H24 ensures that hourly solar
irradiance is never a negative value.
Cells H29 to H32 show that there are four possible values for the fraction of
diffuse solar component (i.e., f1, f2, f3 and f4—not related to those section 3.4.6.1).
Cell H34 chooses the correct value depending on the ratio of total solar irradiance to
extraterrestrial solar irradiance using nested IF functions.
3.4.7 Simulation exercise
3.5
Photosynthesis worksheet
The photosynthesis model component will be implemented in a separate
worksheet. Insert a new worksheet and rename it to Photosynthesis. As before, we
will begin by specifying the model inputs relevant to photosynthesis in this
worksheet:
Row
1
2
3
4
Col. A
INPUT
foliage temp.
LAI
LAI max
Col. B
Cell name
20
1
4
B2: Tf
Eq. ref.
B3: lai
B4: Lmax
The three inputs to photosynthesis are foliage temperature (cell B2), leaf area
index or LAI (cell B3), and maximum LAI (LAI for full canopy closure) (cell B4).
Foliage temperature and LAI will later be simulated when we implement the energy
balance and growth model components, respectively. For now, the first two
parameters will be supplied to the model as constants. Also required are
photosynthesis parameters, some of which are dependent on foliage or canopy
temperature.
101
The following are model parameters that must be adjusted for foliage
temperature:
Row
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Col. A
Tempdependent
MichaelisMenten CO2
MichaelisMenten O2
CO2-O2
specificity
Rubisco
capacity
Non tempdependent
quantum
efficiency
ambient O2
ambient CO2
intercell
CO2
PAR
reflection
PAR
absorption
PAR
scattering
Col. B
value(25)
Col. C
Q10
Col. D
value(Tf)
Cell name
Eq. ref.
300
2.1
D8: Kc
(2.41)
300000
1.2
D9: Ko
(2.42)
2600
0.57
200
2.4
=B8*C8^((Tf25)/10)
=B9*C9^((Tf25)/10)
=B10*C10^((T
f-25)/10)
=(B11*C11^((
Tf-25)/10))
/
(1+EXP(0.29*
(Tf-40)))
(2.43)
D11: Vcmax
0.08
D14: em
210000
350
=0.7*B16
D15: Oa
0.04
D18:
PARreflect
0.8
D19:
PARabsorb
0.8
D20:
PARscatter
(2.45)
D17: Ci
Parameters sensitive to temperature are the Michaelis-Menten coefficient for
CO2 and O2 (cells B8 and B9, respectively), CO2-O2 specificity factor (cell B10), and
Rubisco capacity (cell B11). Their values in cells B8 to B11 are that for foliage
temperature 25 C, and cells D8 to D11 show their corrected values for the given
foliage temperature, Tf. In contrast, cells B14 to B20 are parameters not sensitive to
foliage temperature.
3.5.1 PAR above the canopies
The PAR (photosynthetically active radiation) above the canopies are
calculated as shown below:
Row
1
2
3
4
5
6
7
8
Col. F
PAR above
canopies
PAR direct
PAR diffuse
Col. G
Cell name
Eq. ref.
=0.5*Idr*4.55
=0.5*Idf*4.55
G2: PARdf
(2.29)
(2.30)
k(direct)
taub
1-taub
zenith clump
factor
=0.5/COS(suninc)
=IF(lai<Lmax,1-lai/Lmax,0)
=1-G6
=-LN(G6+G7*EXP(G5*lai/G7))/(G5*lai)
102
G3: PARdf
(2.31)
(2.34)
(2.33)
(2.33)
clump factor
k(direct)
adjusted
=G8^(1-2*suninc/PI())
=G5*G9
11
12
n1
13
n2
14
k(diffuse)
adjusted
=0.6936+13.291*lai^22.3648*lai^4+0.1701*lai^6
=1+22.6522*lai^24.0257*lai^4+0.2674*lai^6
=G12/G13
9
10
15
16
17
LAI sunlit
LAI shaded
=(1-EXP(-kdr*lai))/kdr
=lai-Lsl
G10: kdr
(2.32)
(2.25),
(2.26)
(2.35)
(2.35)
G14: kdf
(2.35)
G16: Lsl
(2.38)
(2.39)
G17: Lsh
The canopy extinction coefficients for direct and diffuse PAR are calculated
(cells G10 and G14, respectively), followed by the sunlit and shaded LAI (cells G16
and G17, respectively). Note that n1 and n2 (cells G16 and G17, respectively) are
intermediary variables.
3.5.2 PAR within the canopies
The PAR within the canopies are next calculated as shown below:
Row
19
20
21
22
23
24
Col. F
PAR within
canopies
direct with
scattering
direct without
scattering
scattering
n
mean diffuse
Col. G
Cell name
=(1-PARreflect)*PARdr*
EXP(-kdr *
SQRT(PARscatter)*lai)
=(1-PARreflect)
*PARdr*EXP(-kdr*lai)
=(G20-G21)/2
=kdf*SQRT(PARscatter)*lai
=((1-PARreflect)*PARdf*(1EXP(-G23)))/G23
Eq. ref.
(2.25)
(2.26)
(2.27)
(2.28)
(2.28)
Recall that there are four PAR components within the canopies: unintercepted
direct PAR with scattering (cell G20), unintercepted direct PAR without scattering
(cell G21), the scattering component only (G22), and the mean diffuse PAR (cell
G24).
3.5.3 PAR absorption
The PAR absorbed by sunlit and shaded leaves are calculated as shown below:
Row
26
27
Col. F
PAR absorbed
sunlit leaves
28
shaded leaves
Col. G
Cell name
Eq. ref.
=PARabsorb *
(kdr*PARdr+G24+G22)
=PARabsorb * (G24+G22)
G27: PARsl
(2.36)
G28: PARsh
(2.37)
3.5.4 CO2 assimilation by leaf and canopy
After calculating the PAR irradiance above and within the canopies, and how
much PAR has been absorbed by the leaves, the next step is to determine the amount
of CO2 assimilated by the leaves. This amount is then scaled up to determine the
canopy assimilation of CO2. The following shows the entries in the spreadsheet:
103
Row
30
Col. F
CO2
assimilation
CO2
compensation
point
Col. G
=Oa/(2*D10)
(2.44)
Rubiscolimited
Sink-limited
Light-limited
(sunlit)
Light-limited
(shaded)
=Vcmax*(Ci-G31) /
(Kc*(1+Oa/Ko)+Ci)
=0.5*Vcmax
=em*PARabsorb*PARsl *
((Ci-G31)/(Ci+2*G31))
=em*PARabsorb*PARsh *
((Ci-G31)/(Ci+2*G31))
(2.47)
37
38
Canopy assim.
=Lsl*MIN(G33,G34,G35) +
Lsh*MIN(G33,G34,G36)
(2.50)
39
itg out
40
Daily canopy
assim. (g CH2O
m-2)
31
32
33
34
35
36
Cell name
=G39*3600*30*10^-6
Eq. ref.
(2.49)
(2.48)
(2.48)
G40: assim
section
2.2.8
section
2.2.8
Cell G38 contains the canopy CO2 assimilation rate for the current local solar
hour, but to determine the daily photosynthetic rate, we have to integrate cell G38
from sunrise to sunset. To do this, we will use BuildIt’s integration operation, ITG.
In the Control worksheet, do the following:

In cell D1, type OPERATIONS.

In cell D2, type daily photosynthesis.

In cell E2, define the name _operation. This indicates to BuildIt that the list of
operations (such as ITG) will be stated starting from cell E2.

In cells E2 to J2, type the following:
2
E
itg
F
=th
G
=Photosynthesis!G38
H
=Photosynthesis!G39
I
=tsr
J
=tss
Recall that ITG has the following specification:
ITG (x, func, output, lower, upper, single_integral)
which specifies the integration of the function func of variable x on the interval
[lower, upper]. The result of integration is to be stored in the output cell. The last
argument, single_integral, specifies if it is a single or multiple integration (TRUE
for the former, FALSE for the latter).
In the case here, argument x (cell F2) is the local solar hour (th), func (cell
G2) is the canopy CO2 assimilation rate (cell G38 in the Photosynthesis worksheet),
output (cell H2) is the integration result to be stored in cell G39 in the
Photosynthesis worksheet, and the integration limits, lower (cell I2) and upper (cell
J2), are sunrise (tsr) and sunset (tss), respectively.
In the Photosynthesis worksheet, cell G40 converts the integration result in
cell G39 to unit g CH2O m-2 by the multiplication of 30x10-6. Multiplication by 3600
is to convert the canopy assimilation rate to unit hour.
104
3.5.5 Simulation exercise
3.6
Energy balance worksheet
Insert a new worksheet and rename it to EnergyBal. This is the worksheet
where the energy balance model component will be implemented. Recall that we had
earlier assumed that the foliage temperature, Tf, was constant when we were building
the photosynthesis model component. In cell B2 in the Photosynthesis worksheet, we
fixed the foliage temperature and defined the name Tf to cell B2. In the EnergyBal
worksheet, however, Tf will be calculated based on the energy balance for a given
solar hour. Consequently, we need to remove the name definition Tf from cell B2 in
the Photosynthesis worksheet.

In Excel 2003’s main menu, choose the commands Insert|Name|Define. In the
dialog box, find the name Tf from the list. Once found, select the name and click
the Delete button to delete it.

In Excel 2007, you have to choose the Name Manager command from the
Formulas ribbon. A dialog box will be shown to display all the names in the
workbook. Find the name Tf from the list. Select the Tf name and click the Delete
button to delete it.
Once deleted, several cells in the Photosynthesis worksheet will display the
error message #NAME? because these cells use the missing name Tf (which we have
deleted). This problem will be rectified later when we define the name Tf to another
cell in the EnergyBal worksheet that holds the calculated foliage temperature.
The energy balance component requires many inputs to be entered, as shown
below:
Row
1
2
Col. B
Cell name
2
B2: zr
1
2
6
Col. A
INPUT
reference
height
crop height
wind ext.
coef.
eddy ext.
coef.
leaf width
7
8
critical LAI
stomata, a1
=Lmax
11.255
9
stomata, a2
0.005
B3: h
B4:
windattn
B5:
eddyattn
B6:
leafwidth
B7: Lcr
B8:
stom_a1
B9:
stom_a2
10
cloud
(longwave)
surface albedo
0.2
3
4
5
11
12
13
2
0.08
0.15
Soil
parameters
105
Eq. ref.
14
15
16
17
18
19
20
water content
saturated
water content
total porosity
pore-size
distribution
tortuosity
dry layer
thickness
roughness
length
0.2
0.45
B15:
vwcsat
0.42
0.18
2
0.02
0.004
B20: zs0
Cell B2 is the height of meteorological measurements, often 2 m or more. The
plant height is specified in cell B3. When we later implement the growth model
component, the plant height will vary according to the simulation time. But as for
now, the plant height is fixed as a constant value. Both the extinction coefficients for
wind speed and eddy diffusivity are fixed at 2 (cells B4 and B5, respectively). Leaf
width is also assumed constant and is specified in cell B6. The critical leaf area index
(cell B7) is needed to scale up the leaf stomatal resistance to the canopy level, and is
assumed to be equal to maximum LAI (cell B4 in the Photosynthesis worksheet).
Both the coefficients for stomatal resistance, a1 and a2, are specified in cells B8 and
B9, respectively. Both describe how stomatal resistance vary according to PAR
irradiance. Cell B10 is the coefficient to correct for cloud cover in the calculation of
the net radiation. The fraction of reflected solar irradiance is fixed at 0.15 (cell B11).
Parameters related to the soil properties are specified in cells B14 to B20.
When the soil water model component is implemented, the soil water content in the
top soil layer will vary according to the various water fluxes in the soil. Until then,
however, we will assume that the volumetric soil water content is constant at 0.2 m 3
m-3 (cell B14). This parameter, along with others (saturated water content, porosity,
tortuosity, dry layer thickness) are needed to calculate the soil surface resistance to
heat flux, and they are each assumed as constant. Cell B20 measures the roughness
length for a tilled, flat soil surface, taken as 4 mm.
Constants such as Stefan-Boltzmann constant and von Karman’s constant are
also specified below:
Row
22
23
Col. A
CONSTANTS
psychometry
Col. B
Cell name
0.658
24
von Karman
0.4
25
air vol. heat
capacity
total solar
scattering
StefanBoltzmann
1221.09
B23:
psycho
B24:
karman
B25: pcp
26
27
0.5
=5.67*10^-8
Eq. ref.
B26:
solarscat
B27: sfb
3.6.1 Net radiation and available energy
After the model parameters and constants are specified, calculations can
resume by determining the net radiation and energy supplied to the system which are
as shown below:
106
Row
1
2
3
4
Col. D
NET RADIATION
Rlu
Rld
RnL
5
6
7
net radiation
8
9
10
11
penetration
probability
soil heat flux
energy to crop
energy to soil
energy to
system
Col. E
Cell name
=sfb*tak^4
=(9.35*10^-6)*sfb*tak^6
=(E3-E2)*(B10+(1-B10) *
sunhr/dl)
=(1-B11)*It+E4
=EXP(-kdr *
SQRT(solarscat)*lai)
=0.35*(E7*Rn)*COS(suninc)
=MAX(0,(1-E7)*Rn)
=E7*Rn-G
=Ac+As
Eq. ref.
(2.54)
(2.55)
(2.52)
E5: Rn
(2.51)
(2.59)
E8: G
E9: Ac
E10: As
E11: A
(2.60)
(2.57)
(2.58)
(2.56)
The MAX function is used in cell E9 to ensure that the energy available to the
crop is always zero or more. Without it, energy available to the crop can be negative
for hours before sunrise or after sunset. Other values that have to be calculated are
related to the air vapour pressure.
3.6.2 Air vapor pressure and wind speed profile
The slope of the saturated air vapor pressure curve (against air temperature)
are calculated next. The difference between the saturated and current air vapor
pressure is known as the vapor pressure deficit. Below shows their determination in
the spreadsheet:
Row
13
14
15
Col. D
VAP. PRESSURE
slope of svp
to temp
vapour
pressure
deficit
Col. E
Cell name
Eq. ref.
=(25029.4*EXP(17.269*ta/(t
a+237.3)))/(ta+237.3)^2
=es-ea
E14: slope
(2.22)
E15: vpd
(2.24)
Model variables related to the wind speed profile are calculated next:
Row
17
18
19
20
21
22
Col. D
WIND SPEED
PROFILE
zero
displacement
hgt
crop roughness
length
friction
velocity
wind speed at
canopy top
eddy
diffusivity at
canopy top
Col. E
Cell name
Eq. ref.
=0.64*h
E18: d
(2.61)
=0.26*(h-d)
E19: z0
(2.62)
=karman*wind/LN(1/0.26)
E20: ustar
(2.63)
=(ustar/karman)*LN((h-d) /
z0)
=karman*ustar*h
3.6.3 Resistances
We can now proceed to calculate the various flux resistances:
107
(2.64)
(2.66)
Row
24
25
Col. D
RESISTANCES
n1
26
27
28
29
30
n2
n3
rsa
31
32
n2
n3
33
34
35
raa
36
37
38
39
40
n2
rca
41
rcs
42
43
44
n1
n1
total PAR
rst
dry soil
resistance
rss
Col. E
Cell name
=(h*EXP(eddyattn)) /
(eddyattn*E22)
=EXP(-eddyattn*zs0/h)
=EXP(-eddyattn*(z0+d)/h)
=E25*(E26-E27)
=LN((zr-d)/(h-d)) /
(karman*ustar)
=h/(eddyattn*E22)
=EXP(eddyattn * (1 (z0+d)/h))-1
=E30+E31*E32
=0.012*lai*(1 EXP(-windattn/2))
=SQRT(E21/leafwidth)
=windattn/(E35*E36)
=IF(It>0,It*0.5,0.01)
=(stom_a1+E39) /
(stom_a2*E39)
=IF(lai<0.5*Lcr, rst/lai,
rst/(0.5*Lcr))
=(B18*B19)/(B16*24.7 *
10^-6)
=E43*EXP(-(1/B17) *
B14/vwcsat)
Eq. ref.
(2.67)
E28: rsa
(2.67)
(2.67)
(2.67)
(2.68)
(2.68)
(2.68)
E33: raa
(2.68)
(2.69)
E37: rca
(2.69)
(2.69)
E40: rst
(2.70)
(2.70)
E41: rcs
(2.71)
(2.72)
E44: rss
(2.73)
Cell E39 calculates the PAR irradiance which is half (50%) of the total solar
irradiance, It. However, a check by the IF function is required that It is not zero
because the PAR irradiance is required by cell E40 to determine the stomatal
resistance, else division by zero occurs. If It is zero, PAR irradiance is set to a very
low value which will give a very high stomatal resistance in cell E40.
3.6.4 Latent heat fluxes
Once the various resistances have been calculated, the total latent heat flux of
the system can finally be calculated in cell H13:
Row
1
2
3
Col. G
ET
Ra
Rc
4
Rs
5
Cc
6
Cs
7
n1
Col. H
Cell name
=(slope+psycho)*raa
=(slope+psycho)*rca +
psycho*rcs
=(slope+psycho)*rsa +
psycho*rss
=1/(1+(H3*H2)/(H4 *
(H3+H2)))
=1/(1+(H4*H2)/(H3 *
(H4+H2)))
=slope*A+(pcp*vpdslope*rca*As)/(raa+rca)
108
Eq. ref.
(2.79)
(2.80)
(2.81)
(2.77)
(2.78)
(2.75)
8
n2
9
10
PMc
n1
11
n2
12
13
14
15
PMs
total ET
=slope+psycho*(1 +
rcs/(raa+rca))
=H7/H8
=slope*A+(pcp*vpdslope*rsa*Ac)/(raa+rsa)
=slope+psycho*(1+rss /
(raa+rsa))
=H10/H11
=H5*H9+H6*H12
vap. pres.
def. at mcf
=vpd+(raa/pcp)*(slope*A(slope+psycho)*LET)
(2.75)
(2.75)
(2.76)
(2.76)
H13: LET
(2.76)
(2.74)
H15: vpd0
(2.82)
After the total latent heat flux has been calculated, we can next calculate the
vapor pressure deficit at the mean canopy flow, D0 (cell H15). D0 is needed to
calculate the latent heat and sensible heat fluxes of the soil and crop components, as
shown below:
+Ro
w
17
18
19
Col. G
Col. H
20
21
ET soil
itg ET soil
22
LATENT HEAT
n1
n2
Cell name
=slope*As+pcp*vpd0/rsa
=slope+psycho *
(rss+rsa)/rsa
=H18/H19
H20: LETs
Daily ET soil
(mm)
=H21*3600/2454000
H22: PETs
23
24
25
n1
n2
26
27
ET crop
itg ET crop
=slope*Ac+pcp*vpd0/rca
=slope+psycho *
(rcs+rca)/rca
=H24/H25
H26: LETc
28
Daily ET crop
(mm)
=H27*3600/2454000
H28: PETc
Daily total ET
(mm)
=H22+H28
29
30
Eq. ref.
(2.83)
(2.83)
(2.83)
section
2.3.10
section
2.3.10
(2.84)
(2.84)
(2.84)
section
2.3.10
section
2.3.10
To determine the amount of water lost from the crop and soil on a daily basis,
we need to integrate the latent heat fluxes over the whole day (from 0.0 to 24.0
hours). As before, we will use BuildIt’s integration operation, ITG.
In the Control worksheet, do the following:

In cells D3 and D4, type daily soil latent heat flux and daily crop
latent heat flux, respectively.

In cells E3 to J4, type the following:
3
4
E
itg
itg
F
=th
=th
G
=LETs
=LETc
H
=EnergyBal!H21
=EnergyBal!H27
109
I
0
0
J
24
24
For convenience, the specification of ITG will be repeated below:
ITG (x, func, output, lower, upper, single_integral)
which specifies the integration of the function func of variable x on the interval
[lower, upper]. The result of integration is to be stored in the output cell. The last
argument, single_integral, specifies if it is a single or multiple integration (TRUE
for the former, FALSE for the latter).
Row 3 in the Control worksheet specifies the integration operation to
determine the daily soil heat flux. The soil heat flux function (LETs or cell H20 in
EnergyBal worksheet) of local solar hour (th or cell B10 in the Control worksheet)
will be integrated from 0 to 24 hours, and the result stored in cell H21 in the
EnergyBal worksheet. Likewise, row 4 in the Control worksheet specifies the
integration of the crop heat flux function (LETc) over the same interval, with the result
being stored in H27 in the EnergyBal worksheet. In the EnergyBal worksheet, the
integration results in cells H21 and H27 are converted to unit mm of water, and they
are stored in cells H22 and H28, respectively.
3.6.5 Sensible heat fluxes and foliage temperature
The daily soil and crop sensible heat fluxes are calculated in a similar manner
as shown below:
Row
32
33
Col. G
SENSIBLE HEAT
n1
34
n2
35
36
H soil
itg H soil
37
Daily H soil
(MJ m-2)
=H36*3600/1000000
38
39
n1
40
n2
41
42
H crop
itg H crop
=psycho*Ac*(rcs+rca)pcp*vpd0
=slope*rca+psycho *
(rcs+rca)
=H39/H40
43
Daily H crop
(MJ m-2)
=H42*3600/1000000
H total
Daily H total
(MJ m-2)
=H35+H41
=H37+H43
foliage temp.
=((H41*rca+H45*raa)/pcp) +
ta
44
45
46
47
48
Col. H
Cell name
=psycho*As*(rss+rsa)pcp*vpd0
=slope*rsa+psycho *
(rss+rsa)
=H33/H34
In the Control worksheet, do the following:
110
Eq. ref.
(2.85)
(2.85)
(2.85)
section
2.3.10
section
2.3.10
(2.86)
(2.86)
(2.86)
section
2.3.10
section
2.3.10
H48: Tf
(2.40)

In cells D5 and D6, type daily soil sensible heat flux and daily crop
sensible heat flux, respectively.

From cells E5 to J6, type the following:
5
6
E
itg
itg
F
=th
=th
G
=EnergyBal!H35
=EnergyBal!H41
H
=EnergyBal!H36
=EnergyBal!H42
I
0
0
J
24
24
Row 5 and 6 in the Control worksheet specify that the functions for soil and
crop sensible heat fluxes (cells H35 and H41 in the EnergyBal worksheet,
respectively) are integrated on the interval [0, 24] hours to obtain their daily values
which are stored, in the EnergyBal worksheet, in cells H36 and H42, respectively.
Cells H37 and H43 in the EnergyBal worksheet store the results in units MJ m-2.
Recall that earlier we had removed the definition of Tf from cell B2 in the
Photosynthesis worksheet. We did that because we will calculate the foliage
temperature from the crop and total sensible heat fluxes and the result will be stored
in cell H48 in the EnergyBal worksheet. Consequently, now define the name Tf to the
cell H48 in the EnergyBal worksheet. All cell formulas containing references to Tf
will now refer to this cell H48.
3.6.6 Simulation exercise
3.7
Tables worksheet (soil water data)
Insert a new worksheet and rename it to Tables. This is the worksheet where
tabulated data are stored. In the next section, we will implement the soil water balance
model component. Consequently, we will store the data related to that component in
this worksheet. Two data sets are required at the moment. They depict the relationship
between: 1) the soil matric suction head and the volumetric water content, and 2) the
hydraulic conductivity and volumetric water content.
The following shows the two tables in the Tables worksheet:
Row
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Col. A
WATER
RETENTION
vwc
0.005
0.01
0.025
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
1
Col. B
Col. D
HYDRAULIC
COND.
vwc
0.005
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
1
matric (m)
10000
3500
1000
200
40
10
6
3.5
2.2
1.4
0.56
0
0
111
Col. E
k (m/day)
3.46E-09
4.32E-08
1.3E-07
6.91E-07
4.32E-06
2.59E-05
0.000173
0.001037
0.006912
0.0432
0.0432
The tabulated data showing the soil water retention characteristics ranges from
A3:B15, whereas the data set for hydraulic conductivity is from D3:E13.
3.8
Soil water balance worksheet
Insert a new worksheet and rename it to WaterBal. This worksheet implements
the soil water balance. As before, we will begin by specifying the inputs for this
model component which are mostly related to soil properties:
Row
1
2
3
4
5
6
7
8
Col. A
INPUT
vwc airdry
water
vwc wilting
point
initial vwc
rooting depth
Col. B
Cell name
0.01
B2: vwcres
0.07
B3: vwcpwp
iteration per
time step
100
fraction
=stepsize/B6
0.4
0.2
Eq. ref.
B5:
rootdepth
B8: frac
The volumetric soil water content at air dry and wilting points are specified in
cells B2 and B3, respectively. Recall that the saturated water content has already been
specified in cell B15 in the EnergyBal worksheet. We are assuming that the soil
profile is uniformly wet at every beginning of the simulation run. The initial soil
water content is specified in cell B4. Cell B5 is the rooting depth, assumed as
constant. When we later implement the plant growth model component, this value will
change according to the simulation time.
The calculations in this worksheet are highly repetitive and time-consuming.
This is because hydraulic conductivity is very sensitive to soil water content.
Choosing a time step size that is too large (e.g., 1 day time step) causes large and
unrealistic fluctuations in the daily soil water content. Consequently, we require that a
within each time step, another iteration cycle be carried out to simulate the soil water
flow. The number of sub-iterations to run for each time step is specified in cell B6. In
the case here, the value given is 100 which is adequate because the hydraulic
conductivity in this example is slow (maximum of 0.0432 m day-1; see the Tables
worksheet). For higher values of hydraulic conductivity, a higher value than 100 subiterations (e.g., 500 or more) may be required to prevent large fluctuations in the daily
water cycle.
Cell B8 contains the formula
=_stepsize/B6
which divides the time step size, _stepsize, by the number of sub-iterations (cell
B6). Recall that for each time step, there will be another iteration cycle to simulate the
soil water flow. The value in cell B8 will be later used to calculate the rainfall
intensity and evaporation rate.
We are now ready to determine the actual evapotranspiration based on the
actual soil water content:
Row
1
Col. D
ACTUAL
Col. E
Cell name
112
Eq. ref.
2
3
4
5
6
EVAPORATION
relative vwc
ETs reduction
AETs (m/day)
=E25/vwcsat
=1/(1+(3.6073*E2)^-9.3172)
=frac*PETs*E3/1000
(2.90)
(2.90)
(2.90)
(2.93)
7
ACTUAL
TRANSPIRATION
critical pt.
8
9
vwc root zone
ETc reduction
10
11
12
AETc (m/day)
=vwcpwp+0.5*(vwcsatvwcpwp)
=MAX(E34:J34)/rootdepth
=IF(E8>=E7,1,
IF(E8<=vwcpwp,0,(E8vwcpwp)/(E7-vwcpwp)))
=frac*PETc*E9/1000
rainfall (m)
=(rain/1000)*frac
(2.92)
(2.92)
E10: AETc
(2.91)
The rates for rainfall (cell E12) and actual evapotranspiration (cells E4 and
E10) are divided by 1000 to convert their values from unit mm to m. They are also
multiplied by frac to adjust their rates due to the sub-iteration cycle within each time
step. Cell E8 contains the calculated volumetric water content in the root zone. Its
calculation methodology which will be explained later.
3.8.1 Partitioning of soil profile into layers and their initial soil water content
The soil profile is divided into six layers, where the first two layers have a
thickness of 0.05 and 0.1 m, respectively. The thicknesses of the other four soil layers
are equal to each other at 0.2 m. As stated earlier, the soil water content in all layers
are assumed equal to each other at the start of every simulation run. The initial
volumetric water content for all layers is specified in cell B4, and their soil water
content, converted to unit mm, are stored in cells E22:J22. Nevertheless, cells E24:J24
are the soil water content at the current time step.
This means that when the simulation begins, the initial soil water content in
cells E22:J22 must be copied to cells E24:J24 to indicate the soil water content for the
layers for the first time step. And to copy these values, BuildIt’s operation, INI, is
used.
In the Control worksheet, do the following:

In cell D20, type PRERUN.

In cell D21, type initial soil water content.

In cell D22, define the name _prerun. This indicates to BuildIt that the list of
initiation operations (such as INI) will be stated starting from cell D22.

In cells E21 to G21, type the following:
21
E
ini
F
=WaterBal!E24:J24
Recall that INI has the following specification:
INI (destination, source)
113
G
=WaterBal!E22:J22
which specifies that the contents from source cells will be copied to the cells referred
to in the destination cells. In our case, cells E22:J22 will be copied to cells E24:J24
(both source and destination cells are in the WaterBal worksheet).
3.8.2 Water fluxes
The main goal of all the calculations in the soil water model component is to
determine the net water flux in each soil layer (cells E42:J42) so that the water
content in the next time step is simply computed as
i,t 1  i,t  qˆi
where i,t and i,t+1 are the water content in soil layer i (m) at time step t and t+1,
respectively; and qˆi is the net flux in soil layer i (m day-1). Cells E24:J24 and E43:J43
are the soil water content for the soil layers at the current time step (i.e., i, t ) and the
next time step (i.e., i, t 1 ), respectively.
For instance, cells E24:J24 and E43:J43 are the water content at time t1 and t2,
respectively. Once the soil water content for all layers are determined at time t2, we
copy the values from cells E43:J43 to cells E24:J24, so that cells E24:J24 and
E43:J43 will now become the water content at time t2 and t3, respectively. We then
repeat these operations until simulation ends. To copy contents from one or more cells
to another, we use BuildIt’s REP operation.
In the Control worksheet, do the following:

In cell D7, type daily water content.

In cells E7 to H7, type the following:
7
E
rep
F
=WaterBal!E24:J24
G
=WaterBal!E43:J43
H
=WaterBal!B6
Recall that REP has the following specification:
REP (destination, source, num_of_replacements)
which specifies that the contents from source cells will be copied to the cells referred
to in the destination cells, and this copying operation will be done
num_of_replacements number of times. In our case, cells E43:J43 will be copied to
cells E24:J24 (both source and destination cells are in the WaterBal worksheet) a
hundred times (cell B6 in the WaterBal worksheet).
Based on the current volumetric water content (cells E25:J25), the hydraulic
conductivity (cells E26:J26) and matric suction head (cells E27:J27) for all soil layers
will be determined by looking up the tabulated data in the Tables worksheet. Looking
up the tables in this case uses the BuildIt’s function, interpolate, which has the
following specification:
interpolate (array_x, array_y, find_x)
Based on an array of given (x, y) values, the interpolate function returns y
corresponding to the given find_x. If no exact match is found, this function performs
a linear interpolation to obtain y. For example,
interpolate(Tables!$A$4:$A$16, Tables!$B$4:$B$16, 0.2)
114
looks up the water retention table in the Tables worksheet (where cells A4:A16 is the
x values and cells B4:B16 the y values) and returns the exact value of 6. However, the
following
interpolate(Tables!$A$4:$A$16, Tables!$B$4:$B$16, 0.22)
returns 5 by linear interpolating between (0.2, 6) and (0.25, 3.5).
Note that the hydraulic conductivities for all layers (cells E26:J26) are
multiplied by frac because there will be several iteration steps (100, in our case)
within each time step to simulate the water flow.
The next step is to determine the hydraulic gradient (which is simply the sum
of the both the matric suction and gravity heads, where the latter is equal to the depth
of the layer from the soil surface) and the mean hydraulic conductivity (cells E28:J28
and E29:J29, respectively). However, before we can calculate the water flux into each
layer, we have to determine the amount of water lost from each layer via evaporation
and transpiration.
Water is lost by evaporation from the first soil layer only because this layer is
in contact with the atmosphere (cells E39:J39). That this layer is in contact with the
atmosphere also means that this layer receives water input via rainfall. Consequently,
the water flux into the first soil layer is equal to the amount of rain it receives minus
water lost by the actual evaporation and transpiration rates (cell E41). For the other
soil layers, water flux is as determined by Darcy’s law and this calculated amount is
additionally subtracted for water lost via transpiration (evaporation is taken as 0 for
all soil layers below the first layer) (cells F41:J41).
Once the water flux into each layer is calculated, the next step is to determine
the net water flux (cells E42:J42) which is simply the difference of the amount of
water flowing into and out of a given soil layer. A positive net flux means the given
soil layer is gaining more water; hence, becoming wetter. In contrast, a negative net
flux means the given soil layer is losing more water and becoming drier.
Cells E43:J43 simply computes the water content for each soil layer for the
next time step. BuildIt will copy their values into cells E24:J24 using the REP
operation which we have specified in the Control worksheet. All the above
calculations repeat, using the soil water content in cells E24:J24 to determine the new
soil water content in cells E43:J43, and the latter values are copied again into cells
E24:J24, and this cycle repeats until the subiteration loop terminates (after a hundred
times in our case).
Finally, we no longer require the volumetric soil water input in cell B14 in the
EnergyBal worksheet to calculate the soil surface resistance to heat fluxes. Instead we
will use the soil water content for the first soil layer as calculated in the WaterBal
worksheet. This means that, in the EnergyBal worksheet, modify the formula in cell
E44 from
=E43*EXP(-(1/B17)*B14/vwcsat)
to
=E43*EXP(-(1/B17)*WaterBal!E25/vwcsat)
115
The following shows the entries in the WaterBal worksheet to determine the net flux of water in the individual soil layers:
Row
14
15
16
17
18
19
20
21
22
23
24
25
26
Col. D
thickness
cum.
thickness
depth
flow distance
airdry water
saturated
water
initial water
soil water
(t1)
vwc
hydraulic
cond.
27
matric
suction
28
hydraulic
gradient
Col. E
SOIL LAYERS
1
0.05
=E16
Col. F
Col. G
Col. H
Col. I
Col. J
Eq. ref.
2
0.1
=F16+E17
3
0.2
=G16+F17
4
0.2
=H16+G17
5
0.2
=I16+H17
6
0.2
=J16+I17
(2.88)
=0.5*E16
=E18
=vwcres*E16
=vwcsat*E16
=E18+0.5*
(E16+F16)
=F18-E18
=vwcres*F16
=vwcsat*F16
=F18+0.5*
(F16+G16)
=G18-F18
=vwcres*G16
=vwcsat*G16
=G18+0.5*
(G16+H16)
=H18-G18
=vwcres*H16
=vwcsat*H16
=H18+0.5*
(H16+I16)
=I18-H18
=vwcres*I16
=vwcsat*I16
=I18+0.5*
(I16+J16)
=J18-I18
=vwcres*J16
=vwcsat*J16
=$B$4*E16
=$B$4*F16
=$B$4*G16
=$B$4*H16
=$B$4*I16
=$B$4*J16
(2.87)
(2.102)
section
2.4.11
=E24/E16
=interpolate
(Tables!
$D$3:$D$13,
Tables!
$E$3:$E$13,
E25)*frac
=interpolate
(Tables!
$A$3:$A$15,T
ables!
$B$3:$B$15,
E25)
=E27+E18
=F24/F16
=interpolate
(Tables!
$D$3:$D$13,
Tables!
$E$3:$E$13,
F25)*frac
=interpolate
(Tables!
$A$3:$A$15,T
ables!
$B$3:$B$15,
F25)
=F27+F18
=G24/G16
=interpolate
(Tables!
$D$3:$D$13,
Tables!
$E$3:$E$13,
G25)*frac
=interpolate
(Tables!
$A$3:$A$15,
Tables!
$B$3:$B$15,
G25)
=G27+G18
116
=H24/H16
=interpolate
(Tables!
$D$3:$D$13,T
ables!
$E$3:$E$13,
H25)*frac
=interpolate
(Tables!
$A$3:$A$15,
Tables!
$B$3:$B$15,
H25)
=H27+H18
=I24/I16
=interpolate
(Tables!
$D$3:$D$13,T
ables!
$E$3:$E$13,
I25)*frac
=interpolate
(Tables!
$A$3:$A$15,
Tables!
$B$3:$B$15,
I25)
=I27+I18
=J24/J16
=interpolate
(Tables!
$D$3:$D$13,T
ables!
$E$3:$E$13,
J25)*frac
=interpolate
(Tables!
$A$3:$A$15,
Tables!
$B$3:$B$15,
J25)
=J27+J18
section
2.4.6
section
2.4.7
(2.101)
29
30
31
mean
hydraulic
cond.
rootdepth/cum
. thickness
n-th layer
=rootdepth/
E17
=AND(E31<=1,
E31>=0)
33
soil water
for roots
34
cum. soil
water (root
zone)
=IF(E32,E25
*(E16-(E17rootdepth)),
E25*E16)
=E33
32
35
36
cj
37
psi
38
transpiration
=MIN(1,E17/
rootdepth)
=1.8*E360.8*E36^2
=E37*AETc
39
40
41
evaporation
=E4
flux in
=E12(E39+E38)
42
43
net flux
soil water
(t2)
=E41-F41
=MAX(E20,
MIN(E21,
E24+E42))
=(E26*E16 +
F26*F16) /
(F16+E16)
=(F26*F16 +
G26*G16) /
(G16+F16)
=(G26*G16 +
H26*H16) /
(H16+G16)
=(H26*H16 +
I26*I16) /
(I16+H16)
=(I26*I16 +
J26*J16) /
(J16+I16)
(2.103)
=rootdepth/
F17
=AND(F31<=1,
F31 >=
(E17/F17))
=IF(F32,F25*
(F16-(F17rootdepth)),
F25*F16)
=IF(AND(
NOT(E32),
E34>0),E34+
F33,0)
=rootdepth/
G17
=AND(G31<=1,
G31 >=
(F17/G17))
=IF(G32,G25*
(G16-(G17rootdepth)),
G25*G16)
=IF(AND(
NOT(F32),
F34>0),F34+
G33,0)
=rootdepth/
H17
=AND(H31<=1,
H31 >=
(G17/H17))
=IF(H32,H25*
(H16-(H17rootdepth)),
H25*H16)
=IF(AND(
NOT(G32),
G34>0),G34+
H33,0)
=rootdepth/
I17
=AND(I31<=1,
I31 >=
(H17/I17))
=IF(I32,I25*
(I16-(I17rootdepth)),
I25*I16)
=IF(AND(
NOT(H32),
H34>0),H34+
I33,0)
=rootdepth/
J17
=AND(J31<=1,
J31 >=
(I17/J17))
=IF(J32,J25*
(J16-(J17rootdepth)),
J25*J16)
=IF(AND(
NOT(I32),
I34>0),I34+
J33,0)
(2.96)
=MIN(1,F17/
rootdepth)
=1.8*F360.8*F36^2
=(F37-E37) *
AETc
0
=MIN(1,G17/
rootdepth)
=1.8*G360.8*G36^2
=(G37-F37) *
AETc
0
=MIN(1,H17/
rootdepth)
=1.8*H360.8*H36^2
=(H37-G37) *
AETc
0
=MIN(1,I17/
rootdepth)
=1.8*I360.8*I36^2
=(I37-H37) *
AETc
0
=MIN(1,J17/
rootdepth)
=1.8*J360.8*J36^2
=(J37-I37) *
AETc
0
(2.99)
=((F28-E28)
*F29/F19)(F39+F38)
=F41-G41
=MAX(F20,
MIN(F21,
F24+F42))
=((G28-F28)
*G29/G19)(G39+G38)
=G41-H41
=MAX(G20,
MIN(G21,
G24+G42))
=((H28-G28)
*H29/H19)(H39+H38)
=H41-I41
=MAX(H20,
MIN(H21,
H24+H42))
=((I28-H28)
*I29/I19)(I39+I38)
=I41-J41
=MAX(I20,
MIN(I21,
I24+I42))
=((J28-I28)
*J29/J19)(J39+J38)
=J41-J26
=MAX(J20,
MIN(J21,
J24+J42))
(2.102)
117
(2.96)
(2.95)
(2.94)
(2.98)
(2.97)
(2.89)
(2.104)
(2.105),
section
2.4.11
3.8.3 Simulation exercise
3.9
Tables worksheet (plant growth data)
Data pertaining to plant growth will be tabulated in the Tables worksheet (the
same location as the tabulated data for soil water). Three types of tables will be stored
here. They are related to the: 1) partitioning of dry matter to various plant parts (such
as stem, leaves, roots and storage organs), 2) specific leaf area, and 3) growth
development rate
Below shows the various tables for dry matter partitioning to the plant parts:
Row
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
Col. A
DRY MATTER
PARTITIONING
(DVS vs.
fraction)
LEAVES
0
0.1
0.25
0.5
0.7
0.95
2.5
Col. B
0.325
0.325
0.462
0.435
0.1395
0
0
STEM
0
0.1
0.25
0.5
0.7
0.95
0.175
0.175
0.198
0.435
0.7905
0.975
1.05
2.5
0
0
Row
38
Col. A
ROOTS
Col. B
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
0
0.1
0.2
0.35
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
2.5
0.5
0.5
0.4
0.22
0.17
0.13
0.1
0.07
0.05
0.03
0.02
0.01
0
0
STORAGE
ORGANS
0
0.95
1.05
2.5
0
0
0.98
0
55
56
57
58
The table for dry matter partitioning to leaves is located in cells A20:B26,
stem in cells A29:B36, roots in cells A39:B52, and storage organs in cells A55:B58.
The partitioning of the dry matter to the various plant parts depends on the growth
development stage (DVS). The partitioning to leaves, for example, is 32.5% from
DVS 0 to 0.1, and increases to 46.2% at DVS 0.25. However, at near anthesis (DVS
0.95), the dry matter partitioning to leaves is zero.
The next tabulated data required is the change in specific leaf area with DVS:
Row
18
19
20
21
Col. F
SPECIFIC LEAF AREA
DVS
0
2.5
118
Col. G
SLA
0.022
0.022
In our case, we are assuming that the specific leaf area is always constant at 0.022 m2
g-1.
The final tabulated data required is the relationship between air temperature
and growth development rate (DVR). Two tables are presented in our case: one for
before anthesis (DVS1) and another for after anthesis (DVS>1):
Row
24
25
26
27
28
29
30
31
32
33
34
Col. F
TEMP vs. GROWTH DEV. RATE
before anthesis
0
30
42
TEMP vs. GROWTH DEV. RATE
after anthesis
0
30
42
Col. G
0
0.01
0.01
0
0.013
0.013
The table for before anthesis is located in cells F26:G28, and for after anthesis in cells
F32:G34.
3.10 Plant growth worksheet
Before we proceed to implement the plant growth component of the model, we
need to remove some name definitions from several cells. Recall that leaf area index,
plant height and rooting depth were previously supplied as constant model inputs
because the plant growth model has yet to be implemented. However, with the plant
growth model component, these model inputs should no longer be required as they
will be calculated and they will vary with the plant growth development stage.

In the Photosynthesis worksheet, remove the name definition lai from cell B3.

In the EnegyBal worksheet, remove the name definition h from cell B3.

In the WaterBal worksheet, remove the name definition rootdepth from cell B5.
After you remove these name definitions, some cells will display the error
message NAME#? because these cells refer to names that we had deleted. For the
moment, ignore these error messages because we will later define the same names to
other cells in the worksheet implementing the plant growth model component.
We are now ready to implement the plant growth component. Insert a new
worksheet and rename it to Growth. For model inputs, we will arrange them as usual
in the upper left corner of the worksheet:
Row
1
2
3
4
5
Col. A
INPUT
max rooting
depth
root growth
rate
max plant hgt
hgt growth
slope
Col. B
Cell name
1
0.012
0.6
0.007
119
Eq. ref.
hgt growth
intercept
base temp
dvs<1
base temp
dvs>1
max dead leaf
coef.
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
67
0
0
0.03
CONSTANTS
Maintenance
stem
leaves
roots
storage organs
0.015
0.03
0.015
0.01
Growth
stem
leaves
roots
storage organs
1.513
1.463
1.444
1.415
Cells B2:B9 are model inputs, where cells B2 and B3 denote the maximum
rooting depth and root growth rate, respectively. Cells B5 and B6 are the coefficients
for plant height growth, and cells B7 and B8 are the base temperatures for plant
growth before and after anthesis, respectively (both taken as 0 C). Lastly, cell B9 is
the coefficient for maximum leaf death.
Cells B13:B16 are the coefficients for maintenance requirement for the
various plant parts. Likewise, cells B19:B22 are the coefficients for growth
requirement for the plant parts.
3.10.1 Growth development rate and stage
The scale for the plant growth development stage (DVS) ranges from 0 (plant
emergence) to 1 (flowering) and ends with 2 (maturity). DVS is incremented based on
the current growth development rate (DVR) as shown below:
Row
2
3
4
5
6
Col. D
dev. stage
(DVS) (t1)
dev. rate
(DVR) preanthesis
dev. rate
(DVR) postanthesis
DVR
Col. E
Cell name
E2: dvs
dev. stage
(DVS) (t2)
=dvs+E5*_stepsize
=interpolate(Tables!
F26:F28, Tables!
G26:G28,tmean)
=interpolate(Tables!
F32:F34, Tables!
G32:G34,tmean)
=IF(dvs<=1,E3,E4)
Eq. ref.
(2.106)
section
2.5.1
section
2.5.1
section
2.5.1
(2.106)
Cell E2 is the current growth development stage (DVS) and its next DVS is
determined in cell E6 which contains the formula
=dvs+E5*_stepsize
120
which can be represented symbolically as
s,t 1  s,t  r ,t t
where s,t and s,t+1 are the DVS at time t and subsequent time step t+1; t is the time
step size (days); r,t is the growth development rate (DVR) (days-1) at time t. The
plant’s growth development rate, as stated earlier, depends on air temperature. This
relationship is tabulated in the Tables worksheet earlier, where there are two sets of
table: one for before and another for after anthesis. using the interpolate function,
the DVR before anthesis is contained in cell E3, and the DVR after anthesis is in cell
E4. In cell E5, DVR is set equal to cell E3 if DVS is greater than or equal to 1;
otherwise DVR is set equal to E4 for DVS > 1. Using this current value for DVR, the
DVS for the next time step is calculated in cell E6. Once calculated, the value in cell
E6 is copied to cell E2, and the cycle repeats. To implement this copying operation,
BuildIt’s REP is used (see section 3.10.6).
Also note that cell E2 is left blank because, as stated earlier, BuildIt will copy
the value of DVS in cell E6 to cell E2 using the REP operation. However, we must
supply an initial value of DVS for the first time step. This initial value will be
supplied using BuildIt’s INI operation (see section 3.10.6).
3.10.2 Maintenance and growth respiration
Water stress effects the amount of assimilates actually produced. It is reduced
linearly depending on the ratio between the actual and potential transpiration rates.
This ratio was previously calculated in cell E9 in the WaterBal worksheet. Below
shows their calculations in Excel:
Row
8
9
Col. D
Growth
reduction
Assimilates
Col. E
=WaterBal!E9
=assim*E8
Cell name
Eq. ref.
(2.121)
(2.122)
Cell E8 merely refers to cell E9 in the WaterBal worksheet, and the actual
assimilates available for maintenance and growth is scaled accordingly in cell E9 in
the Growth worksheet.
Cells E13:E17 are the current weights of the plant parts, and they are used to
determine their maintenance requirement (cells F13:F17). The maintenance
requirement for all part parts are then summed (cell F18) and adjusted for air
temperature (cell E21) and plant age (cell E22). Cell E23 ensures that the total
maintenance requirement does not exceed the total available assimilates. Assimilates
left after fulfilling the maintenance requirement is subsequently available for growth
(cell E24).
Growth requirement for individual plant parts (cells H13:H17) are calculated
based on the dry matter partitioning to the plant parts (cell G13:G17) which are
obtained from the tabulated data in the Tables worksheet using the interpolate
function. The growth requirement for all the plant parts are summed (cell H18) and
used to determine the growth rate of the individual plant parts (cells E25 and I13:I17).
Once the growth rates are calculate, we can update the weights of the plant parts for
the next time step.
121
As stated earlier, cells E13:E17 are the current weights of the plant parts, and
their weights are updated (for the next time step) in cells J13:J17 which contain the
formula, described symbolically as
Wi,t 1  Wi, t  i t
where Wi,t and Wi,t+1 are the weights (g m-2 ground area) of a given plant part i (stem,
roots or storage organs) at time step t and subsequent time step t+1, respectively; t is
the time step size (days); and i is the growth rate for plant part i. This translates to
mean that weights Wi,t and Wi,t+1 are cells E13:E17 and J13:J17, respectively; growth
rates, i, are cells I13:I17; and time step size, t, is _stepsize.
Note that the weight gain by leaves are calculated differently than other plant
parts because we have to account for leaf death, if any. Leaf death can be caused by
two possibilities: 1) due to leaf age (cell E28), or 2) due to self-shading (cell E29).
The maximum of those two is the actual leaf death (cell E30). This value is then used
in cell I15 to calculate the death rate of leaves. Consequently, the difference between
the growth and death rates of the green leaves determine the weight gain or loss of the
green leaves (cell J14). The MAX function in cell J14 ensures that weight of green
leaves do not become negative.
Once the weights of the plant parts are updated in J13:J17, their values are
copied to cells E13:E17. These calculations repeat to determine the weight change in
the plant parts for every subsequent time step. To enable copying of cells J13:J17 to
E13:E17, BuildIt’s REP operation is used (see section 3.10.6). Also note that cells
E13:E17 are blank because their values will be copied from cells J13:J17, as
described earlier. However, we must supply the initial weights to these plant parts for
the first time step. For this purpose, we will use BuildIt’s INI operation (see section
3.10.6).
122
The following illustrates the calculation of the various plant growth processes in the Growth worksheet:
Row
11
Col. D
Current plant
weight
Col. E
Col. F
Col. G
Col. H
Col. I
Col. J
weight (t1)
partitioning
=interpolate
(Tables!
A29:A36,
Tables!
B29:B36,dvs)
=interpolate
(Tables!
A20:A26,
Tables!
B20:B26,dvs)
growth req.
=G13*B19
growth rate
=G13*E25
weight (t2)
=E13+I13*
_stepsize
=G14*B20
=G14*E25
=MAX(0,E14+
(I14-I15)*
_stepsize)
(2.109),
(2.110),
(2.116)
=E14*E30
=E15+I15*
_stepsize
=E16+I16*
_stepsize
(2.115)
=E17+I17*
_stepsize
(2.109),
(2.110),
(2.111)
12
13
stem
maint. req.
=E13*B13
14
green leaves
=E14*B14
15
dead leaves
16
roots
=E16*B15
17
storage
organs
=E17*B16
18
total
19
20
Respiration
=SUM(E13:E17)
=SUM(F13:
F17)
=interpolate
(Tables!
A39:A52,
Tables!
B39:B52,dvs)
=interpolate
(Tables!
A55:A58,
Tables!
B55:B58,dvs)
=SUM(G13:
G17)
123
=G16*B21
=G16*E25
=G17*B22
=G17*E25
=SUM(H13:
H17)
=SUM(J13:
J17)
Eq. ref.
(2.109),
(2.110),
(2.111)
(2.109),
(2.110),
(2.111)
21
maint (tempadjust)
22
pot. maint
23
24
total maint
available for
growth
total growth
25
26
27
28
Leaf death
due to old
age
29
due to selfshading
30
actual leaf
death rate
=F18*2^
((tmean-25)
/10)
=E21*E14/
SUM(E14:E15)
=MIN(E22,E9)
=E9-E23
(2.109)
(2.109)
(2.109)
(2.111),
(2.116)
(2.111),
(2.116)
=IF(H18>0,
E24/H18,0)
=IF(E2>1,
interpolate(
Tables!
F32:F34,
Tables!
G32:G34,
tmean)/
MIN(0.1, 2dvs),0)
=IF(lai>Lcr,
MIN(B9,
B9*(lai-Lcr)
/Lcr),0)
=MAX(E28,E29)
(2.113)
(2.114)
(2.112)
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3.10.3 Leaf area index
Calculations to determine the leaf area index (LAI) is simple (cell E34). It is
merely the weight of the green leaves (cell E14) multiplied by the specific leaf area
(cell E33), where the latter is obtained from the tabulated data in the Tables
worksheet. The following shows their determination:
Row
32
33
Col. D
LAI
specific leaf
area
34
LAI
Col. E
Cell name
=interpolate(Tables!
F20:F21,Tables!G20:G21,
dvs)
=E14*E33
Eq. ref.
(2.117)
E34: lai
(2.117)
Note that the name lai is defined for cell E34.
3.10.4 Rooting depth
Roots only grow when three conditions are met: 1) DVS is before anthesis
(cell E38), 2) soil water content in the root zone (cell E8 in the WaterBal worksheet)
is above the soil’s permenant wilting point (cell E39), and 3) the current rooting depth
is less than the maximum rooting depth (cell E40). When all these three conditions are
true, the rooting depth is updated for the next time step (cell E42). The value in cell
E42 is then copied to cell E36 so that these calculations can be repeated to determine
the growth of the rooting depth for every subsequent time step. To copy cell E42 to
E36, we will use BuildIt’s REP operation (see section 3.10.6). We also need to specify
the initial rooting depth for cell E36 using BuildIt’s INI operation (see section
3.10.6).
The following shows the calculations for the rooting depth:
Row
36
37
38
39
40
41
42
Col. D
Rooting depth
rooting depth
(t1)
pre-anthesis?
water >
wilting point?
less than max
depth?
root growth
rate
rooting depth
(t2)
Col. E
Cell name
Eq. ref.
E33:
rootdepth
(2.120)
=dvs<1
=WaterBal!E8>vwcpwp
(2.120)
(2.120)
=E37<B2
(2.120)
=B3*E8
(2.120)
=IF(AND(E38,E39,E40),E37+
E41*stepsize,E37)
(2.120)
Note that the name rootdepth is defined for cell E37.
3.10.5 Plant height
The growth of plant height requires the calculation of the daily degree day
(cell E46) and cumulative degree days (cell E47). To sum the daily degree day in cell
E46 to E47, we will use BuildIt’s ACC operation (see section 3.10.6). We next proceed
to calculate the plant growth rate (cell E52) to determine the plant height for the next
time step (cell E53). This value is then copied to cell E49 so that these calculations
are repeated to determine the plant height growth for every subsequent time step. To
copy cell E53 to E49, we will use BuildIt’s REP operation (see section x3.10.6). As
125
before, we also need to supply an initial value for plant height in cell E49 for the first
time step. To do this, BuildIt’s INI operation is used (see section 3.10.6).
The following shows the calculations for plant height growth:
Row
44
45
46
47
48
49
50
51
52
53
Col. D
Plant height
base temp
degree day
cum. degree
days
plant height
(t1)
n1
n2
height growth
rate
plant height
(t2)
Col. E
Cell name
=IF(dvs<=1,B7,B8)
=MAX(0,tmean-E45)
Eq. ref.
(2.107)
(2.107)
(2.108)
E49: h
(2.118)
=B6*B5*B4*EXP(-B5*E47)
=1+B6*EXP(-B5*E47)
=(E46*E50/E51^2)*E8
(2.119)
(2.119)
(2.119)
=E49+E52*_stepsize
(2.118)
Note that the name h is defined for cell E49.
3.10.6 BuildIt’s operations for plant growth
The model component for plant growth requires several BuildIt’s operations
(INI, REP and ACC) to accomplish its goals. Initial values must be supplied to: 1) DVS,
2) rooting depth, 3) plant height, and 4) the individual plant part weights.
In cells D22 to G29 in the Control worksheet (in the _prerun section), type
the following:
22
23
24
25
26
27
28
29
D
DVS
rooting depth
plant height
stem wgt
green leaves wgt
dead leaves wgt
roots wgt
storage organs wgt
E
ini
ini
ini
ini
ini
ini
ini
ini
F
=dvs
=rootdepth
=h
=Growth!E13
=Growth!E14
=Growth!E15
=Growth!E16
=Growth!E17
G
0
0.03
0.01
0.3
0.5
0
0.8
0
This means that the initial value for DVS is 0 (denoting plant emergence), rooting
depth is 0.03 m and plant height is 0.01 m. Recall that the weights for the plant parts
are located in cells E13:E17 in the Growth worksheet. The initial values for stem,
green leaves, dead leaves, roots and storage organs are 0.3, 0.5, 0.0, 0.8 and 0.0 g,
respectively. At the start of a simulation run, BuildIt will refer to the _prerun section
in the Control worksheet and copy the values in column G to their specified
destinations as specified in column F.
In cells D8 to H12 in the Control worksheet (in the _operation section), type
the following:
8
9
D
degree days
daily plant part weights
E
acc
rep
F
=Growth!E46
=Growth!E13:E17
126
G
=Growth!E47
=Growth!J13:J17
H
+
1
10
11
12
rooting depth
plant height
growth dev. stage
rep
rep
rep
=rootdepth
=h
=dvs
=Growth!E42
=Growth!E53
=Growth!E6
1
1
1
Rows 9 to 12 indicate that the values from the source cells as specified in column G
will be copied to the destination cells as specified in column F. For example, in the
Growth worksheet, values from cells J13:J17 will copied to E13:E17, from E42 to
E33 (rootdepth), from E53 to E49 (h), and from E6 to E2 (dvs). These copy
operations are to ensure that the plant part weights, rooting depth, plant height and
DVS are updated for the next time step. The value 1 in column H indicate that these
copy operations should only occur once per time step.
In row 8, the ACC operation is used to accumulate the daily degree day to
determine the cumulative degree days. Recall that the ACC operation has the following
specification:
ACC (x, accumulated_x, operator)
where operator can either be + or * for addition or multiplication, respectively. If the
operator is addition (as in our case), ACC is equivalent to
accumulated_x 
 xi
i
But if the operator is multiplication, ACC is equivalent to
accumulated_x 
 xi
i
Cells E8:H8 specify to BuildIt that it should sum each time step’s daily degree day
from cell E46 and place the total in cell E47 in the same Growth worksheet.
It is important to note that ACC operation (row 8) should be specified before
the REP operation for updating the plant height (row 11). This is because the growth of
plant height uses the cumulative degree days in its calculations.
3.11 Finalizing the model
3.11.1 Criteria to end simulation run (Control worksheet)
There are two more tasks left to do to complete building the gcg model in
Excel. The first is to modify the criteria to stop the model simulation run. Previously,
we stated that simulation should end after a specified number of runs; that is, in the
Control worksheet, we entered the following criteria in cell B5:
=(step/_stepsize)<maxsteps
We will add one more criterion. Simulation run should continue so long as: 1) the
specified number of runs has not been reached, and 2) that the growth development
stage of the plant (DVS) is 2 or less. Consequently, in cell B5 in the Control
worksheet, enter the following:
=AND((_step/_stepsize)<maxsteps,dvs<=2)
Recall that the AND function returns TRUE only when both conditions
(_step/_stepsize)<maxsteps and dvs<=2 are TRUE.
127
3.11.2 Model output results (output worksheet)
The second task to do before we complete building the model is to specify
which model variables should be displayed at every time step. Model output is
displayed in the Output worksheet. In the Output worksheet, we previously defined
the name _read to cell A2. BuildIt uses this cell position as a starting point to
determine which model variables should be displayed at every time step; that is, all
cells to its right will be read by BuildIt to determine which model variables are to be
displayed.
Complementing _read is _write because the latter name determines where
the model results should be displayed. We previously defined cell A6 in the Output
worksheet with the name _write. Consequently, this cell position will be used as a
starting position where the output will be displayed; that is, all cells to its right will be
used for displaying model output.
In the Output worksheet, type the following (due to space constraints, the
following table is presented with rows and columns running vertically and
horizontally, respectively):
Col.
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
Row 1
date
elapsed
rain
vwc layer 1
vwc layer 2
vwc layer 3
vwc layer 4
vwc layer 5
vwc layer 6
DVS
LAI
GreenLv
DeadLv
Stem
Roots
Storage
Hgt
Root depth
Photosyn
ETc red
Assim
Maint
Growth
ETc
ETs
Total ET
Row 2
=date
=date-Control!B7
=rain/1000
=WaterBal!E25
=WaterBal!F25
=WaterBal!G25
=WaterBal!H25
=WaterBal!I25
=WaterBal!J25
=dvs
=lai
=Growth!E14
=Growth!E15
=Growth!E13
=Growth!E16
=Growth!E17
=h
=rootdepth
=assim
=Growth!E8
=Growth!E9
=Growth!E23
=Growth!E25
=PETc
=PETs
=EnergyBal!H30
Row 3
FALSE
FALSE
FALSE
FALSE
FALSE
FALSE
FALSE
FALSE
FALSE
FALSE
FALSE
FALSE
FALSE
FALSE
Cell A2 is the simulation date, and cell B2 is the number of elapsed days since
the simulation start date. Cell C2 is the amount of rainfall (in meters), and is a
convenient reference variable because the following six cells (D2:I2) are the
volumetric soil water content for the first to the sixth and last layer. Cell J2 is the
growth development stage of the plant, followed by cell K2 which is the leaf area
128
index. Cells L2:P2 are the weights for the green leaves, dead leaves, stem, roots and
storage organs, respectively. Plant height is contained in cell Q2, followed by the
rooting depth in cell R2. The total amount of assimilates produced potentially is in
cell S2. However, due to water stress, this amount can be reduced by a fraction as
displayed in cell T2 to give the actual assimilates produced via photosynthesis in cell
U2. Cells V2 and W2 are the maintenance and growth respiration rates, respectively.
the last three cells concern the potential rates for evaporation, transpiration and their
total.
The FALSE values in some cells in the third row are to instruct BuildIt to
output the values of the corresponding cells at the beginning of every iteration run
(Fig. xxx). If left blank or if the cells contain TRUE, BuildIt will only display the
values of the corresponding cells at the end of each iteration run.
How do we decide if to display a given variable at the start or end of every
iteration run? We should choose the former option (i.e., FALSE) if we wish to display
the variable’s value before it is changed in the iteration cycle. For example, in the first
time step, t1, we supplied initial values to the soil water content for each soil layer. In
this case, the values of the soil water content before entering the loop should be
displayed. This is because when the model enters the first iteration cycle, the water
fluxes are calculated and the soil water content in each layer is updated for the second
time step, t2. If we choose to display their values at the end of the iteration cycle (i.e.,
TRUE), we then erroneously display the soil water content at t2 for t1.
This is why other variables such as leaf area index, weights of the plant parts,
plant height and rooting depth should be displayed before each iteration cycle. Their
values will be updated for the next time step within the iteration cycle. Other variables
such as photosynthesis and evapotranspiration should be displayed at the end of each
iteration cycle because their values are only known after they have been calculated
within each iteration cycle.
The variables listed in cells A2:Z2 in the Output worksheet are only examples
of model variables that can be displayed in every iteration cycle. You may add more
variables or delete some of them as required. Once the model output has been
specified, the model can be run by choosing the Start Simulation from the BuildIt
menu.
129