James W. Jones

Wheat yield functions for analysis of land-use change in China

CERES-Wheat, a dynamic process crop growth model, is specified and validated for eight sites in the major wheat-growing regions of China. Crop model results are then used to test the Mitscherlich-Baule and the quadratic functional forms for yield response to nitrogen fertilizer, irrigation water, temperature, and precipitation. The resulting functions are designed to be used in a linked biophysical-economic model of land-use and land-cover change in China. While both functions predict yield responses adequately, the Mitscherlich-Baule function is preferable to the quadratic function because its parameters are biologically and physically realistic. Variables explaining a significant proportion of simulated yield variance are nitrogen, irrigation water, and precipitation; temperature was a less significant component of yield variation within the range of observed year-to-year variability at the study sites. Crop model simulations with a generic soil with median characteristics of the eight sites compared to simulations with site-specific soils showed that agricultural soils in China have similar and, in general, low-to-moderate water-holding capacities and organic matter contents. The validated crop model is useful for simulating the range of conditions under which wheat is grown in China, and provides the means to estimate production functions when experimental field data are not available.

Basics of Agricultural System Models

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. In this chapter, we present concepts of system models with examples, along with methods for developing system models. System models can be viewed in two different, yet complementary ways. First, a model can be treated as a system of differential or difference equations that describe the dynamics of the system. Second, the model can be treated as a set of static equations that describe how responses of interest at particular times depend on explanatory variables. We present and discuss these two viewpoints in this chapter. As we shall see, the different methods described in this book may call for one or the other of these viewpoints. In this chapter, we start by presenting general systems concepts and definitions that are needed for modeling systems. Then we go through the process of developing a model of a system using an example with two simple interacting components that will help give students an intuitive understanding of the system modeling process. Example system models are presented for several important agricultural system components, demonstrating some of the key features and relationships used in model development.

Putting It All Together in a Case Study

In the previous chapters, methods for working with dynamic system models were presented using various examples, chosen to illustrate the principles of the methods. What is missing there is the way that the methods interact within a single modeling project. In fact, there is to a large extent a logical progression in a modeling project, from an exploration of the data, to a preliminary test of the model, to uncertainty analysis and sensitivity analysis, to model calibration, then to another round of evaluation, and finally to application of the model. In many cases, one step uses information from the previous steps. It is this progression and interaction that we illustrate in this chapter. Doing a case study also requires us to choose, when several different approaches are possible. This may also be helpful, since each modeler will be faced with similar choices. The case study uses the simple maize model, and has as its objective to map maize yield and inter-annual variability over Europe. Only the most important parts of the R code to apply the methods are shown. The main results are shown and discussed. All the steps can be easily re-run using demonstration R scripts (demos) in the R package ZeBook.

Parameter Estimation with Bayesian Methods

Bayesian methods are becoming increasingly popular for estimating parameters of complex mathematical models because the Bayesian approach provides a coherent framework for dealing with uncertainty. To start with, the principle is a prior probability distribution of the model parameters describing our belief about the parameter values before we use the set of measurements. The Bayesian methods then tell us how to update this belief using the measurements to give the posterior parameter density. In the Bayesian approach, the parameters are defined as random variables and the prior and posterior parameter distributions represent our belief about parameter values before and after using observed data to improve estimates. This approach has several advantages: i) parameters can be estimated from different types of information (data, literature, expert knowledge); ii) the posterior probability distribution can be used to implement uncertainty analysis methods; iii) the posterior probability distribution can be used for optimizing decisions in the face of uncertainty. This chapter presents the basic principles of the Bayesian approach and describes several algorithms to calculate posterior parameter distributions. These algorithms are illustrated in several applications on yield and soil carbon estimation.

Statistical Notions Useful for Modeling

We first define the notion of a random variable, and then the notion of a probability distribution function. Then we consider multiple random variables, and the ways of describing distributions in this case. We are particularly interested in conditional distributions, since a regression model is in fact an approximation to the conditional expectation of the response variable, given the explanatory variables. Next we consider sampling and samples. Since our information about the random variables of interest comes from sampling, it is essential to show how sampling quantities are related to population quantities. We discuss simple random sampling and also sampling schemes more relevant in agronomy. Next we present regression models. The last section is a brief introduction to Bayesian statistics.

The R Programming Language and Software

This chapter is an introduction to R programming language and software to provide the basis for the reader to implement the methods presented in the following chapters. The chapter begins with instructions on how to obtain R and how to take your first steps into this environment. The main R data objects are presented (vectors, matrices, data frames, and lists). Interactions with external data to read them from file or write to file are also discussed. Notions of programming with loops, conditional executions, and functions are particularly important for working with dynamic models. Graphics and statistical capabilities are introduced briefly too. R packages provide functions to be used for applying methods and you will learn how to use these packages. Finally, two topics that are particularly important for our subject and are discussed include: how to use R to interact with a model programmed in a different language and how to reduce computing time.

Data Assimilation for Dynamic Models

In this chapter, we consider modifying a dynamic model using data assimilation methods to make it a better predictor for a specific situation (e.g., for a specific agricultural field). We consider a family of methods often referred to as filtering. A filter is an algorithm that is applied to a time series to improve it in some way (like filtering out noise). Here the time series is the successive values for the model state variables, and the improvement comes from using measured values to update the model state variables. The state variables are updated sequentially, i.e., each time an observation is available. We present several filtering methods developed for linear and non-linear dynamic models. We illustrate these methods in case studies on yield predictions, water balance, soil carbon, and weed population dynamics. We also show how filtering can be implemented with R.

Simulation with Dynamic System Models

Computer simulation is used to produce numerical values of the state variables of a dynamic system model over time. Computer programs are created to encode the mathematical equations that describe the dynamic model, specify initial conditions and environmental inputs, then start a program loop during which the changes to system state variables are computed at each time step and state variables are incremented to produce values for the next time step. Repeating this set of calculations as time progrefsses from a starting to an ending value, one obtains estimated values of the system’s state variables vs. time. A general set of steps are used to simulate dynamic models represented by differential equations in continuous time and those represented by difference equations with time progressing in discrete steps. In this chapter, we present three different numerical methods used to obtain approximate solutions to continuous time models written as ordinary differential equations (Euler, Improved Euler, and Runge-Kutta Fourth Order methods). Errors associated with use of numerical approximations are discussed relative to the importance of selecting appropriate time steps. We also discuss the implementation of computer code in R for simulating difference equation models, pointing out similarities and differences with programs written to simulate continuous time dynamic models. Examples of R programs are presented and discussed for various dynamic models, including a continuous differential equation model of insect population dynamics and difference models for a soil water balance and a simple maize crop model.