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1.1 What if there was life on Mars? 1 1.2 What is a model? 2 1.3 Why use models? 4 1.4 Which model should I use? 8 1.41 Determining what type of model to use 9 1.4.2 Determining what type of mathematics to use in the mode 15 1.5 Choosing an existing model 20 1.6 How is a model made? 21 Summary 23 Problems 24 2 How to develop a model 26 2.1 Choose the type of model 27 2.1.1 Why am I doing this? 27 2.12 How should I do this? 27 2.1.3 The principle of parsimony 29 2.1.4 What do I do now? 30 2.2 Draw up a conceptual model 31 2.2.1 Draw a pictue 32 2.2.2 List all hypotheses 33 2.2.3 List all assumptions 33 2.2.4 Set the boundary conditions 33 2.3 Attach a mathematical model 35 2.3.1 The parts of a mathematical model 36 2.3.2 Linking together fixed parameters and input variables 36 23.3 Choosing the mathematical approach to derive the model 38 2.3.4 Example 40 2.4 Construct a computer model 45 2.4.1 General spreadsheets 46 2.4.2 Specialist modelling software 52 2.4.3 High-level programming languages 55 2.5 And then..? 61 Summary 62 Problems 63 Further reading 65 References 67 Web links 6 How to evaluate a model 69 3.1 Decide what type of evaluation is needed 70 3.2 Plot the results (graphical analysis) 71 3.2.1 Plots to reveal the accuracy of the simulation 72 3.2.2 Plots to illustrate the behaviour of the model components 76 3.23 Plots to establish the important model components 78 3.3 Calculate the accuracy of the simulation (quantitative analysis) 81 "33.1 Analysis of ccincidence 84 3.3.2 Analysis of association 94 3.3.3 An example of the use of statistics to assess the accuracy of a model: the model of soil carbon change on Mars 98 3.4 Examine the behaviour of the model (sensitivity analysis) 106 3.4.1 What is sensitivity analysis and why is it important? 106 3.4.2 Methods used in sensitivity analysis 107 3.43 Expressing sensitivity 107 3.5 Determine the importance of the model components (uncertainty analysis) 109 3.5.1 What is uncertainty analysis and why is it important? 109 3.5.2 Methods used in uncertainty analysis 110 3.5.3 Representing variation in the input parameters and model outputs 110 3.54 Expressing uncertainty 111 3.6 Andthen.. ? 113 Summary 114 Problems 115 Further reading 117 References 117 Web links 118 S How to apply a model 119 4.1 Scientific representation 122 4.11 Who is the end-user? 123 41.2 How is the model used? 124 4.1.3 Guard against input error 126 4.1.4 Guard against misinterpretation of the results 126 4.1.5 Documentation 127 4.2 Expert and decision support systems 128 4.2.1 Who is the end-user? 129 4.2.2 How is the model used? 130 4.2.3 Guard against input error 130 4.2.4 Guard against misinterpretation of the resuts 132 4.2.5 Documentation 132 4.3 Risk assessment 134 4.3.1 Who is the end-user? 135 4.3.2 How is the model used? 136 43 Gua rd aga ins nut error 137 4 .34 Guard against isinterpreation of ithe r ts 37 43.5 Documentation 38 .4 Spatially-explicit applications 139 4.4.1 Who is the end-usr ? 139 4.4.2 How io the model used? 40 4.3 Guard againstnput err 143 4.44 Guard against misinterpretaion oi the O reuls 44 4.4,5 Documentation 144 45 Epilogue 6 4.51 How has life on Mars been improved? 46 4.5.2 The real-Earth application of the models ed in this book 47 4.6 So is this the end? 10 Summary 152 Problems 156 Further readin 157 References 157 Web links 158

Library of Congress subject headings for this publication: Environmental sciences Mathematical models