1 INTRODUCTION .................................................. 1
1.1 OBJECTIVE ................................................ 2
1.2 STRUCTURE ................................................ 2
2 INFORMATION THEORY ............................................ 4
2.1 HOW TO MEASURE INFORMATION LOSS .......................... 4
2.2 HOW TO ESTIMATE INFORMATION LOSS WHEN THE TRUTH IS
UNKNOWN .................................................. 7
3 MODEL SELECTION METHODS ....................................... 9
3.1 MODEL SELECTION METHODS IN THE LITERATURE ................ 9
3.2 PRINCIPLE OF PARSIMONY .................................. 11
3.3 AKAIKE'S INFORMATION CRITERION (AIC, AKAIKE 1973):
MODEL SELECTION BASED ON THE ESTIMATION OF THE
INFORMATION LOSS ........................................ 11
3.4 AIC MODIFICATIONS (AICC, QAIC, QAICC, AND AICC*) ........ 15
3.4.1 Small sample correction: AICC(Hurvich and Tsai
1989) ............................................ 15
3.4.2 Modified criteria for overdispersion: QAIC and
QAICC (Lebreton et al. 1992) ..................... 20
3.4.3 Symmetric criterion: AICC* (Seghouane and Amari
2007) ............................................ 21
3.5 BAYESIAN INFORMATION CRITERION (BIC, SCHWARZ 1978) AND
KASHYAP INFORMATION CRITERION (KIC, KASHYAP 1982) ....... 21
3.6 DISCUSSION ABOUT THE TRUE MODEL NOT BEING PRESENT IN
THE SET OF CANDIDATE MODELS FOR CORRECT DERIVATION OF
AIC AND BIC ............................................. 23
3.7 COMPARISON OF MODEL SELECTION METHODS IN HYDROGEOLOGY ... 25
3.8 IMPLEMENTATION PROBLEMS OF MODEL SELECTION METHODS IN
HYDROGEOLOGY ............................................ 25
4 MODEL ASSESSMENT WITH DIFFERENT TYPES OF OBSERVATIONS USING
SYNTHETIC DATA ............................................... 31
4.1 NORMING DIFFERENT KINDS OF OBSERVATIONS ................. 31
4.2 SYNTHETIC DATA GENERATING MODEL ......................... 33
4.2.1 Conceptual model and boundary conditions ......... 33
4.2.2 Geometry ......................................... 34
4.2.3 Initial heads .................................... 34
4.2.4 Recharge and hydraulic conductivity fields ....... 35
4.2.5 Generation of heterogeneity ...................... 36
4.2.6 Effective porosity ............................... 37
4.2.7 Parametrization of MODFLOW packages: River,
Well and Solver .................................. 38
4.2.8 Transport model .................................. 38
4.2.9 Dispersion ....................................... 39
4.2.10 Concentration source ............................. 40
4.2.11 Time discretization .............................. 40
4.2.12 Numerical stability .............................. 40
4.2.13 Boreholes and observations ....................... 42
4.2.14 Model results .................................... 42
4.3 CANDIDATE MODELS ........................................ 44
4.4 MODEL SELECTION ANALYSIS WITH DIFFERENT TYPES OF
OBSERVATIONS ............................................ 46
4.4.1 Model selection analysis with one type of
observations ..................................... 46
4.4.2 Model selection analysis with two types of
observations ..................................... 48
4.4.3 Model selection analysis with three types of
observations ..................................... 51
4.5 DISCUSSION .............................................. 54
4.5.1 Impact of number of observations ................. 54
4.5.2 Impact of type of observations ................... 58
4.5.3 Impact of parameter order ........................ 59
5 MODEL ASSESSMENT WITH DIFFERENT TYPES OF OBSERVATIONS USING
REAL DATA .................................................... 67
5.1 REAL DATA ............................................... 67
5.2 LYSIMETER MODEL ......................................... 68
5.3 SENSITIVITY ANALYSIS .................................... 75
5.4 MODEL ASSESSMENT ........................................ 77
5.5 DISCUSSION AND CONCLUSION ............................... 79
6 CONCLUSION AND OUTLOOK ....................................... 81
APPENDIX A: BASIC MATHEMATICAL ASPECTS TO UNDERSTAND AIC ........ 84
A.1 ESTIMATING PARAMETERS WITH THE MAXIMUM LIKELIHOOD
ESTIMATION METHOD ....................................... 84
A.2 MAXIMUM LIKELIHOOD ESTIMATE FOR THE NORMAL
DISTRIBUTION CASE ....................................... 90
A.3 ВAYESIAN INFERENCE ...................................... 92
APPENDIX B: MODEL SELECTION ISSUES .............................. 94
B.1 TAKEUCHI'S INFORMATION CRITERION (TIC, TAKEUCHI 1976) ... 94
B.2 MULTIMODEL INFERENCE .................................... 95
B.3 HILL AND TIEDEMAN'S (2007A) METHOD TO OBTAIN THE VALUE
OF AIC FOR DIFFERENT TYPES OF OBSERVATIONS .............. 96
B.4 SYNTHETIC DATA .......................................... 98
B.5 NORMALIZED SYNTHETIC DATA .............................. 100
REFERENCES ..................................................... 101
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