**"Spatial and Temporal Statistics - Sampling Field Soils and their Vegetation"** is intended to introduce concepts and theories to scientists already familiar with classical statistics and one or more disciplines of agricultural science. Each chapter introduces one concept and its application to several sets of field-measured data. Rather to begin each chapter with rigorous assumptions coupled with a comprehensive theory and procedure, we introduce the concept with typical questions that may be answered by applying the concept to a set of observations. Examples of data from various field studies observed by colleagues and ourselves are used as a frame for explaining basic concepts of spatial statistics and how to apply them within and between fields. The original data, the analysis and the interpretation are followed by a discussion of issues and concerns associated with the underlying assumptions of the analysis. At the end of each chapter, the reader can select references that comprehensively describe the theoretical basis of the concept and limitations of its application. Like classical statistics, spatial and temporal statistical methods consist of tools only - no more, no less - and do not provide any miracle capable of replacing the ideas and creativity of the scientist.

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Table of Contens
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**Preface**

**1 Review of descriptive statistics 1**

1.1 Mean and variance 1

1.2 Frequency distribution 2

1.2.1 Histograms 2

1.2.2 Fractile diagrams 3

1.3 Mode, median, skewness and kurtosis 5

1.4 Additional examples of histograms 7

1.4.1 Infiltration rate 7

1.4.2 Surface soil temperature 9

1.4.3 Mineral soil nitrogen 14

1.5 "Outliers" 14

1.6 Covariance 16

1.7 Linear regression 16

1.7.1 Examples of regression 17

1.7.1.1 Soil elevation and distance 17

1.7.1.2 Mineral soil nitrogen 17

1.7.2 Prudence regarding regression 17

1.7.2.1 Soil surface temperature and irrigation water salinity 19

1.7.2.2 Wheat yield and fine textural particles 20

1.7.2.3 Saturated hydraulic conductivity and porosity 22

1.8 References for the theory and calculations 24

1.9 Exercises and problems 25

**2 Autocorrelation 31**

2.1 Relevant questions 31

2.2 Framework for calculation 32

2.3 Results of the analysis for several data sets 35

2.4 Interpretation of the analysis 39

2.4.1 Observation length and correlation length 39

2.4.2 Frequency distribution and spatial or temporal relationships 42

2.5 Precautions and related topics 43

2.5.1 Selection of sampling interval 43

2.5.2 Stationarity and trends 44

2.5.2.1 Deterministic variations 46

2.5.3 Interpolation and contours 49

2.5.4 Minimum size of treated plots 49

2.5.5 Field-measured variables in deterministic equations 50

2.6 Potential research topics 51

2.6.1 Quantifiying taxonomic soil properties 51

2.6.2 Quantifying landscape soil properties 52

2.7 References for the theory and basis of the calculations 53

2.8 Exercises and problems 53

**3 Cross correlation 63**

3.1 Relevant questions 63

3.2 Framework for calculation 63

3.3 Example data sets 64

3.3.1 Soil water retention curve 64

3.3.1.1 Cross correlation results 68

3.3.1.2 Interpretation of the analysis of the soil water retention data 71

3.3.2 Wheat yield, soil water use and nitrogen use 74

3.3.3 Remotely sensed crop nitrogen status 75

3.3.4 Soil water pressure head 75

3.4 Other considerations 80

3.5 Precautions 82

3.5.1 Significance of cross correlation 82

3.6 Potential research topics 83

3.7 References for theory and basis of the calculation 84

3.8 Exercises and problems 85

**4 Semivariograms 91**

4.1 Relevant questions 91

4.2 Framework for calculation 91

4.2.1 Anisotropy 96

4.3 Examples of data sampled on a transect 97

4.3.1 Bounded or transitional semivariograms 97

4.3.2 Unbounded or nontransitional semivariograms 102

4.4 Examples of data sampled on a grid 104

4.5 Interpretation of the analysis 106

4.6 Precautions 109

4.7 Potential research topics 110

4.8 References for theory and basis of the calculations 111

4.9 Exercises and problems 112

**5 Kriging 119**

5.1 Relevant questions 119

5.2 Framework for calculation 120

5.2.1 Illustrative example of the calculation 121

5.3 Examples of different kinds of kriging 124

5.3.1 Punctual kriging along a transect 124

5.3.2 Punctual kriging across a grid 126

5.3.3 Block kriging 129

5.3.3.1 Block kriging along a transect 130

5.3.3.2 Block kriging across an area 133

5.3.4 Indicator kriging 135

5.3.4.1 Indicator kriging of soil water content 136

5.3.4.2 Indicator kriging of dissolved organic carbon 140

5.3.5 Universal kriging 145

5.4 Interpretation of the analysis 149

5.4.1 Cross-validation or jack-knifing 149

5.5 Precautions 153

5.5.1 Concepts of stationarity 153

5.5.2 Semivariograms - always suspect 153

5.5.3 Kriging - a smoothing process 155

5.5.4 Impact of nugget on kriging 155

5.5.5 Meaning of the kriging variance 155

5.6 Potential research topics 156

5.7 References for the theory and basis of the calculation 157

5.8 Exercises and problems 158

**6 Crossvariograms and cokriging 168**

6.1 Relevant questions 168

6.2 Framework for calculations 169

6.2.1 Crossvariogram 169

6.2.2 Cokriging 169

6.3 Example data sets of crossvariograms and cokriging 170

6.3.1 Almond yields 170

6.3.2 Spring barley grain yield and soil water content 173

6.4 Interpretation of the analysis 178

6.5 Precautions 181

6.5.1 Linear model of co-regionalization and the problem of positive definiteness 181

6.5.2 Other precautions 184

6.6 Potential research topics 185

6.7 References for theory and basis of the calculations 188

6.8 Exercises and problems 189

**7 Spectral analysis 196**

7.1 Relevant questions 196

7.2 Framework for calculations 196

7.3 Results of the analysis for several data sets 199

7.3.1 Boron concentration within a soil profile 199

7.3.2 Spatial intercropping patterns 199

7.3.3 Soil temperature 202

7.3.4 Wheat yield, soil water use and soil nitrogen use 203

7.3.5 Surface soil water content at different sampling dates 204

7.4 Potential research topics 205

7.5 Precautions 207

7.6 References for the theory and basis of the calculations 209

7.7 Exercises and problems 210

**8 Cross spectral analysis and coherency 215**

8.1 Relevant questions 215

8.2 Framework for calculation 216

8.3 Results of the analysis for several data sets 221

8.3.1 Soil temperature and irrigation water quality 221

8.3.2 Hourly microbial respiration, air temperature and soil temperature 223

8.3.3 Daily microbial respiration, soil temperature and rainfall 225

8.3.4 Temporal variation of water stored within soil profiles 229

8.3.5 Crop yield and soil nitrogen use 230

8.4 Potential research topics 232

8.5 Precautions 234

8.6 References for the theory and basis of the calculations 235

8.7 Exercises and problems 235

**9 Autoregressive and moving average functions 240**

9.1 Relevant questions 240

9.2 Framework for calculation 240

9.2.1 Random walk model 240

9.2.2 Autoregressive model AR(p) 241

9.2.3 Moving average model MA(q) 242

9.2.4 Autoregressive moving average model ARMA(p, q) 242

9.3 Example data sets 242

9.3.1 A random soil water content distribution 242

9.3.2 Linearly increasing elevation 244

9.3.3 Fluctuating boron concentration 245

9.3.4 A second order AR model 246

9.3.5 An ARMA model 248 9.4 Interpretation of the analysis 250

9.5 Precautions 251

9.6 Potential research topics 251

9.6.1 Alternative description of landscape attributes 251

9.6.2 Alternative analysis of landscape attributes 252

9.6.2.1 Sorghum yield 253

9.6.2.2 Bromide distribution 255

9.7. References for theory and basis of the calculations 257

9.8 Exercises and problems 258

**10 Autoregressive state-space analysis 263**

10.1 Relevant questions 265

10.2 Framework for calculation 265

10.2.1 State-space theory for autoregressive models 265

10.2.2 Kalman filtering and EM algorithm 267

10.2.3 Model identification and interpretation 269

10.2.4 First steps in the inspection of state-space analysis 271

10.2.5 Irregular observations 284

10.3 Example data set 287

10.3.1 Surface soil temperature and soil water content 287

10.3.2 Sorghum yield, soil salinity and soil water content 292

10.3.3 Soil salinity and inorganic solutes 294

10.3.4 Wheat grain yield and selected landscape observations 297

10.3.5 Nitrogen fixation 299

10.3.6 Hourly microbial respiration and air temperature 301

10.3.7 Daily microbial respiration, soil temperature and rainfall 302

10.4 Interpretation of the analysis 304

10.5 Precautions 306

10.6 Potential research topics 307

10.7 References for theory and basis of the calculations 308

10.8 Exercises and problems 310

**11 Physical state-space models 316**

11.1 Relevant questions 316

11.1.1 Relating the physical process to the relevant state-space model 316

11.1.2 Obtaining the necessary data to estimate the state-space model 317

11.1.3 Post-analysis questions 317

11.2 Framework for calculations 318

11.2.1 Brief review of physical equations in state-space models 318

11.2.2 State-space theory for physical equations 319

11.3 Applications of physical equations in state-space models 321

11.3.1 Time series of soil temperature 321

11.3.2 Time series of nitrogen mineralization 332

11.3.3 Carbon respiration and associated soil temperature 338

11.3.4 Soil hydraulic property determination and associated experimental uncertainty 341

11.4 Precautions and suggestions 347

11.5 Potential research topics 349

11.5.1 Impact of measurement resolution 349

11.5.2 Underlying simplifying assumptions - a problem? 350

11.5.3 Applications for monitoring land surface processes 352

11.6 References for theory and basis of the calculations 355

**12 Postscript 358**

**13 Appendices 367**

**14 Index 389**