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Working with Dynamic Crop Models
Methods, Tools and Examples for Agriculture and Environment
Working with Dynamic
Crop Models
Methods, Tools and Examples for Agriculture
and Environment
Third Edition
Daniel Wallach
Institut National de Recherche Agronomique (INRA), France
David Makowski
Institut National de Recherche Agronomique (INRA), France
James W. Jones
Agricultural and Biological Engineering Department, University of Florida,
Gainesville, FL, United States
François Brun
Acta, French Agricultural Technical Institutes, France
Academic Press is an imprint of Elsevier
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© 2019 Elsevier B.V. All rights reserved.
No part of this publication may be reproduced or transmitted in any form or by any means, electronic
or mechanical, including photocopying, recording, or any information storage and retrieval system,
without permission in writing from the publisher. Details on how to seek permission, further
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This book and the individual contributions contained in it are protected under copyright by the
Publisher (other than as may be noted herein).
Notices
Knowledge and best practice in this field are constantly changing. As new research and experience
broaden our understanding, changes in research methods, professional practices, or medical
treatment may become necessary.
Practitioners and researchers must always rely on their own experience and knowledge in
evaluating and using any information, methods, compounds, or experiments described herein.
In using such information or methods they should be mindful of their own safety and the safety of
others, including parties for whom they have a professional responsibility.
To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors,
assume any liability for any injury and/or damage to persons or property as a matter of products
liability, negligence or otherwise, or from any use or operation of any methods, products,
instructions, or ideas contained in the material herein.
xiii
xiv Preface
The last section is almost entirely new, treating (with one exception) topics
that were not covered at all in the second edition. This is the section on advanced
methods and new uses of system models. The chapters here treat metamodels
(how to approximate a system model by a simpler model), multimodel ensem-
bles (how to analyze the results from ensembles of models), gene-based model-
ing (how to combine agronomic and genetic information in a model), data
assimilation for dynamic models (how to update a system model using informa-
tion on an individual), and models as an aid to sampling.
This book can be used for self-study, as a reference volume, or as a textbook
for a course in modeling. Examples of the latter include courses at the
University of Florida, at Universidade de Passo Fundo in Brazil, at Nanjing
Agricultural University in China, and multiple short intensive courses around
the world. The chapters are mostly self-contained so that you can read, study,
or draw on the chapters in any order. Each chapter has a series of exercises at the
end to test your understanding and capability of applying the methods.
The principal authors of the chapters were as follows: Basics of agricultural
system models (JWJ), The R programming language and software (FB), Sim-
ulation with dynamic system models (JWJ), Statistical notions useful for
modeling (DW), Regression analysis (DW), Uncertainty and sensitivity analy-
sis (DM), Calibration of system models (DW), Parameter estimation with
Bayesian Methods (DM), Model evaluation (DW), Putting it all together in a
case study (FB), Metamodels (DM), Multimodel ensembles (DW), Gene-based
modeling (JWJ), Data assimilation for dynamic models (DM), and Models as an
aid to sampling (DM).
Acknowledgments
Third edition:
The authors express gratitude to Dr. C.E. Vallejos, Dr. M. Bhakta, Dr. K.J.
Boote, and Dr. M. Correll for providing data used in Chapter 13. The authors
also want to thank Dr. M. Bhakta, Dr. K.J. Boote, Dr. G. Hoogenboom, and
Dr. C.E. Vallejos for reviewing an earlier version of Chapter 13 (Gene-Based
Crop Models), and for making helpful suggestions for improving the chapter.
Second edition:
The authors gratefully acknowledge their home institutions:
l INRA (Institut National de la Recherche Agronomique).
l The University of Florida, Agricultural and Biological Engineering
Department.
l ACTA, head of the network of French Technical Institutes for Agriculture.
The following projects provided an invaluable framework and support for
discussing and applying methods for working with dynamic crop models:
l Reseau Mixte Technologique Modelisation et Agriculture, funded by a grant
from the French Ministry for Agriculture and Fisheries.
l The project “Associate a level of error in predictions of models for agron-
omy and livestock” (2010–13), funded by a grant from the French Ministry
for Agriculture and Fisheries.
l AgMIP, the Agricultural Model Intercomparison and Improvement Project.
l The FACCE-JPI project MACSUR (Modeling European Agriculture with
Climate Change for Food Security).
l The INRA Metaprogramme ACCAF (Adaptation de l’agriculture et de la
for^et au changement climatique).
In addition, we owe thanks to:
l Luc Champolivier, Terres Inovia, for providing field water content
unpublished data.
l Sylvain Toulet, intern at INRA in 2012, Juliette Adrian, intern at ACTA in
2013, and Lucie Michel, intern at ACTA-INRA, for their contributions to
the examples.
xv
xvi Acknowledgments
l Senthold Asseng of the University of Florida, for working with the authors
and making use of the new material in this book in the graduate course,
“Simulation of Agricultural and Biological Systems.”
l All the students in the courses we have given on crop modeling, who have
enriched our thinking with their questions and remarks.
Chapter 1
1 INTRODUCTION
Agricultural systems are complex combinations of various components. These
components contain a number of interacting biological, physical, and chemical
processes that are manipulated by human managers to produce the most basic of
human needs—food, fiber, and energy. Whereas the intensity of management
varies considerably, agricultural production systems are affected by a number
of uncontrolled factors in their environments, being exposed to natural cycles of
weather, soil conditions, pests, and diseases. In comparison to physical and
chemical systems, agricultural systems are much more difficult to manage
and control because of the living system components that respond to their phys-
ical, chemical, and biological environmental conditions in highly nonlinear,
time-varying ways that are frequently difficult to understand. These interactions
and nonlinearities must be considered when attempts are made to model and
predict agricultural systems’ responses to their environments and management.
2 SYSTEM MODELS
A system is a set of components and their interrelationships that are grouped
together by a person or a group of persons for purposes of studying some part
of the real world. Usually, a group of persons works together to define the sys-
tem that they intend to analyze, and in many cases the individuals are from dif-
ferent disciplines because of the scope of the system they intend to study. The
selection of the components to include in a particular system model depends on
the objectives of analysts and on their understanding and perspectives of the real
world. A system may have only a few components or it may have many com-
ponents that interact with each other and that may be affected by factors that are
not included in the system. Conceptually, a system may consist of only one
component; however, in this book we focus on the more common situation that
exists in agricultural systems where the complexities of interactions among
components are required to understand the performance of the overall system
being studied.
FIGURE 1.1 Schematic diagram of a cropping system model showing interactions between soil
and crop components in the system and influences of explanatory variables from the environment. In
this system, it is assumed that weather, management, pests in the biotic environment and soil vari-
ables in the soil environment (e.g., below 1 m in depth) influence the system but are not affected by
system components.
affect weather. A cropping system (Figure 1.1) may include soil and a crop that
interact with each other and with the environment. The environment may have
several variables that define the characteristics of the environment that directly
influence some of the system component processes. The soil and crop system is
usually defined as a homogeneous field or a representative area in the field that is
exposed to weather throughout the course of a season. The boundary of this sys-
tem would be an imaginary box immediately surrounding this representative
uniform area in the field (e.g., with dimensions of 1 ha by 1 ha or of 1 m by
1 m in area), and a lower boundary in the soil at 1 m in depth. The environment
would be everything in the soil and atmosphere outside this imaginary box that
would affect the soil and crop behavior in the system. Note that each system
component may have multiple variables that describe the conditions of system
components at any particular time. For example, the soil component may include
soil water content, mineral nitrogen content, and soil organic matter that change
from day to day. The crop component may include above-ground biomass, leaf
area, and depth of roots in the soil.
This example also illustrates another important point: assumptions have to be
made when choosing a system’s components and its environment. Here it is
assumed that the temperature and humidity in the canopy are equal to the values
of these variables in the air mass above the canopy. In reality, the soil and crop
conditions affect canopy temperature and humidity through the transfer of heat
and water vapor into the air above the crop, and thus the system does affect the
immediate environment of the crop. This interaction may be important in many
situations such that the system would include the canopy temperature and
Basics of Agricultural System Models Chapter 1 7
humidity. In the example, the assumption is that those effects are small relative to
the influences of external factors, such as that of the external air mass. This
assumption is made in most, but not all, existing cropping system models.
This example of a system and its environment can also be used to illustrate
the implications of incorporating additional components into a system. If can-
opy air temperature and humidity are added to the soil and crop components, the
system will include another component, the canopy air environment. This will
cause the model to become more complex in that explicit mathematical relation-
ships would be needed to model the dynamics of canopy air temperature and
humidity in addition to the soil and crop conditions. The environment for this
expanded system would still include weather explanatory variables above the
crop canopy.
2.3.2 Simulation
Simulation refers to the numerical solution of the system model to produce
values of the variables that represent the system components over time. Although
in some literature, a model is referred to as a “simulation model,” this terminol-
ogy hides an important and distinctive difference between a system model and
its solution. Computer code should be developed based on the mathematical
description and explicit assumptions. Otherwise, the assumptions and
8 SECTION A Background
relationships may be hidden in the computer code and not easily understood or
communicated to each modeling team member and to the outside world. Sim-
ulation is discussed in more detail in Chapter 3.
dðU1 ðtÞ
¼ g1 ðUðtÞ, XðtÞ, θÞ
dt
⋮ (1)
dðUS ðtÞ
¼ gS ðUðtÞ, XðtÞ, θÞ
dt
where t is time, U(t) ¼ [U1(t), …, US(t)]T is the vector of state variables (defined
below) at time t, X(t) is the vector of environmental variables at time t (some-
times referred to as exogenous, forcing, or driving variables), θ is the vector of
system component properties that are included in the model to compute rates of
change, and g1, g2, …, gS are some functions. The state variables U(t) could
include, for example, leaf area index (leaf area per unit of soil area), crop
biomass, root depth, soil water content in each of several soil layers, etc. The
explanatory variables X(t) typically include climate variables (such as daily
maximum and minimum temperatures) and management variables (such as
irrigation dates and amounts). As discussed in Section 2.5, in the rest of this
book we will use X(t) and θ to refer to explanatory variables and parameters,
respectively, which is not quite the same definition as above.
The model of Equation (1) is a system of first-order differential equations
that describes rates of changes in each of the state variables of the system. It
is dynamic in the sense that the solution (simulation) of the system of equations
describes how the state variables evolve over time. It describes a system in the
sense that there are several state variables that interact.
Continuous time dynamic systems are sometimes modeled using discrete
time steps; in this case, the models are mathematically represented as a set of
difference equations. We will later show how this is useful in simulation of
the dynamic system model and the interpretation of dynamic models as response
functions. By writing the left side of Equation (1) as a discrete approximation of
the derivative, it is straightforward to develop the general form of a dynamic
system model in discrete time:
There is an important concept underlying the use of Equations (1), (2) to rep-
resent dynamic systems. Note that numerical methods are almost always needed
to solve for behavior of system model variables over time, and Equation (2) is a
highly useful and simple approach for solving dynamic system models.
However, if the intent is to closely approximate the exact mathematical solu-
tion of a dynamic system model (i.e., if it could be solved analytically), one
needs to select the time step Δt such that the numerical errors associated with
approximating continuous time by discrete time steps are acceptable. Opera-
tionally when solving the model, one can evaluate the effects of choosing dif-
ferent values of Δt on important model results. On the other hand, model
developers may select the Δt when developing the model because of the level
of understanding of processes in the system and on the availability of environ-
mental variables. In this case, all the model equations and explanatory vari-
ables have to be developed specifically for the chosen time step, and this
may require different approximations to the processes than those used when
solving a model using continuous time. The mathematical form of such
discrete-time models may be represented as in Equation (2), and such models
were referred to as “functional models” by Addiscot and Wagenet (1985). An
example is modeling soil water flow versus soil depth and over time using
the Richards equation (continuous time model) versus using a so-called
tipping bucket approach (with a daily time step). This will be discussed more
in Chapter 3.
In some agricultural system models, such as in many dynamic crop models,
Δt is 1 day. One reason for this choice of time steps in many crop models is that
highly important weather data may be available only on a daily basis (e.g., daily
rainfall). In this case, the Δt on the left side becomes 1 and it disappears on the
right side of the equation. Note that Equation (2) is an approximation to the con-
tinuous model form. The choice of difference equations to model dynamic sys-
tems may mean that the model developer must use functional relationships that
approximate the underlying physical, chemical, or biological processes
involved. Thus, the development of functions (gi) that compute changes in
the state variables need to take into account the time step used in discrete time
models.
system being studied, then soil mineral nitrogen will be a state variable and the
model will include an equation to describe the evolution over time of this vari-
able. If soil mineral nitrogen is not included as a state variable, it could still be
included as an explanatory variable, that is, its effect on plant growth and devel-
opment could still be considered. However, in this case the values of soil mineral
nitrogen over time would have to be supplied to the model; they would not be
calculated within the model. The limits of the system being modeled are different
in the two cases.
The choice of state variables is also fundamental for a second reason. It is
assumed that the state variables at time t give a description of the system that is
sufficient for calculating the future trajectory of the system. For example, if
only root depth is included among the state variables and not variables describ-
ing root geometry, the implicit assumption is that the evolution of the system
can be calculated on the basis of just root depth.
Furthermore, past values of root depth are not needed. Whatever effect they
have is assumed to be taken into account once one knows all the state variables
at time t. Given a dynamic model in the form of Equation (1) or (2), it is quite
easy to identify the state variables. A state variable is a variable that appears
both on the left side of an equation, so that the value is calculated by the model,
and on the right side because the values of the state variables determine the
future trajectory of the system.
FIGURE 1.5 Forrester diagram of the simple maize crop growth model. The model has three state
variables (TT(t), B(t), and LAI(t)) and three daily weather explanatory variables (TMAX(t), TMIN(t),
and I(t)). See text for definitions of variables.
solar radiation, and crop biomass. However, this approach provides a good first
approximation to LAI development under potential growth conditions.
Figure 1.5 shows the Forrester diagram of this system, showing the three
system state variables, the system boundary, the processes that cause these vari-
ables to change versus time, parameters that were used in the functions used to
compute the rates of change of state variables, and the explanatory weather vari-
ables that must be known in order to simulate crop growth. Note that there are
three rectangular boxes or levels to represent the three state variables. Note also
that the rate of accumulation of thermal time age of the crop, dTT(t), depends
only on daily maximum and minimum temperatures and an explanatory
variable, Tbase (Equation 11). Thermal age of the crop is the accumulation
(integration) of this rate.
TMIN ðtÞ + TMAXðtÞ
dTT ðtÞ ¼ max Tbase; 0 (11)
2
Equation (11) shows that dTT(t) is restricted to nonnegative values. The rate of
leaf area expansion depends on thermal age of the crop in that leaves expand
only during a vegetative phase of crop growth, which is defined in this model
by TTL, the thermal age of the crop when the crop ends its vegetative phase and
begins reproductive growth. The rate of expansion is also reduced as LAI
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Length to end of tail 18 1/2 inches, to end of wings 11 3/8; extent of
wings 22 1/2; wing from flexure 8; tail 10 1/2; bill along the ridge
1 4 1/2/12; tarsus 1 10 1/2/12; first toe 8/12, its claw 10/12; middle toe
1 2/12, its claw 6/12.
Fig. 1.
Fig. 2.
This species extends its range from the mouth of the Columbia
River, across our continent, to the shores of the Gulf of Mexico; but
how far north it may proceed is as yet unknown. On the 10th of April
1837, whilst on Cayo Island, in the Bay of Mexico, I found a
specimen of this bird dead at the door of a deserted house, which
had recently been occupied by some salt-makers. From its freshness
I supposed that it had sought refuge in the house on the preceding
evening, which had been very cold for the season. Birds of several
other species we also found dead on the beaches. The individual
thus met with was emaciated, probably in consequence of a long
journey and scanty fare; but I was not the less pleased with it, as it
afforded me the means of taking measurements of a species not
previously described in full. In my possession are some remarkably
fine skins, from Dr Townsend’s collection, which differ considerably
from the figure given by Bonaparte, who first described the species.
So nearly allied is it to the Green-crested Flycatcher, M. crinita, that
after finding the dead bird, my son and I, seeing many individuals of
that species on the trees about the house mentioned, shot several of
them, supposing them, to be the same. We are indebted to the
lamented Thomas Say for the introduction of the Arkansaw
Flycatcher into our Fauna. Mr Nuttall has supplied me with an
account of its manners.
“We first met with this bold and querulous species, early in July, in
the scanty woods which border the north-west branch of the Platte,
within the range of the Rocky Mountains; and from thence we saw
them to the forests of the Columbia and the Wahlamet, as well as in
all parts of Upper California, to latitude 32°. They are remarkably
noisy and quarrelsome with each other, and in the time of incubation,
like the King Bird, suffer nothing of the bird kind to approach them
without exhibiting their predilection for battle and dispute. About the
middle of June, in the dark swamped forests of the Wahlamet, we
every day heard the discordant clicking warble of this bird, somewhat
like tsh’k, tsh’k, tshivait, sounding almost like the creaking of a rusty
door-hinge, somewhat in the manner of the King Bird, with a
blending of the notes of the Blackbird or Common Grakle. Although I
saw these birds residing in the woods of the Columbia, and near the
St Diego in Upper California, I have not been able to find the nest,
which is probably made in low thickets, where it would be
consequently easily overlooked. In the Rocky Mountains they do not
probably breed before midsummer, as they are still together in noisy
quarrelsome bands until the middle of June.”
Dr Townsend’s notice respecting it is as follows: “This is the Chlow-
ish-pil of the Chinooks. It is numerous along the banks of the Platte,
particularly in the vicinity of trees and bushes. It is found also, though
not so abundantly, across the whole range of the Rocky Mountains;
and among the banks of the Columbia to the ocean, it is a very
common species. Its voice is much more musical than is usual with
birds of its genus, and its motions are remarkably quick and graceful.
Its flight is often long sustained, and like the Common King Bird, with
which it associates, it is frequently seen to rest in the air, maintaining
its position for a considerable time. The males are wonderfully
belligerent, fighting almost constantly, and with great fury, and their
loud notes of anger and defiance remind one strongly of the
discordant grating and creaking of a rusty door hinge. The Indians of
the Columbia accuse him of a propensity to destroy the young, and
eat the eggs of other birds.”
Not having seen this handsome bird alive, I am unable to give you
any account of its habits from my own observation; but I have
pleasure in supplying the deficiency by extracting the following notice
from the “Manual of the Ornithology of the United States and of
Canada,” by my excellent friend Thomas Nuttall.
“This very beautiful and singular species of Flycatcher is confined
wholly to the open plains and scanty forests of the remote south-
western regions beyond the Mississippi, where they, in all probability,
extend their residence to the high plains of Mexico. I found these
birds rather common near the banks of Red River, about the
confluence of the Kiamesha. I again saw them more abundant, near
the Great Salt River of the Arkansa in the month of August, when the
young and old appeared, like our King Birds, assembling together
previously to their departure for the south. They alighted repeatedly
on the tall plants of the prairie, and were probably preying upon the
grasshoppers, which were now abundant. At this time also, they
were wholly silent, and flitted before our path with suspicion and
timidity. A week or two after, we saw them no more, having retired
probably to tropical winter-quarters.
“In the month of May, a pair, which I daily saw for three or four
weeks, had made a nest on the horizontal branch of an elm,
probably twelve or more feet from the ground. I did not examine it
very near, but it appeared externally composed of coarse dry grass.
The female, when first seen, was engaged in sitting, and her mate
wildly attacked every bird which approached their residence. The
harsh chirping note of the male, kept up at intervals, as remarked by
Mr Say, almost resembled the barking of the Prairie Marmot, ’tsh,
’tsh, ’tsh. His flowing kite-like tail, spread or contracted at will while
flying, is a singular trait in his plumage, and rendered him
conspicuously beautiful to the most careless observer.”
Muscicapa forficata Gmel. Linn. Syst. Nat. vol. i. p. 931.—Lath. Ind. Ornith.
vol. ii. p. 485.—Ch. Bonaparte, Synopsis of Birds of United States, p. 275.
Swallow-tailed Flycatcher, Muscicapa forficata, Bonap. Amer. Ornith.
vol. i. p. 15, pl. 2, fig. 1.
Swallow-tailed Flycatcher, Nuttall, Manual, vol. i. p. 275.