Universit´e de Li`ege Facult´e des Sciences Appliqu´ees Array recordings of ambient vibrations: surface-wave inversion A thesis submitted for the degree of Doctor of Applied Sciences presented by Marc Wathelet February, 2005 Contents List of figures iv List of tables ix R´esum´e xi Abstract xiii Introduction 1 Objectives . 2 Thesis outline . 3 1 Measuring wave velocity 5 1.1 Ambient vibrations . 5 1.1.1 Frequency-wavenumber method . 6 1.1.2 High resolution method . 12 1.1.3 Spatial auto-correlation method . 13 1.2 Artificial sources . 14 1.2.1 P S refracted waves . 15 − V 1.2.2 SH refracted waves . 17 1.2.3 Surface wave inversion . 17 1.3 Conclusions . 18 2 The inversion algorithm 19 2.1 Definition . 19 2.2 Available methods . 20 2.2.1 Gridding method . 21 2.2.2 Iterative methods . 21 2.2.3 Neural Networks . 22 2.2.4 Monte Carlo methods . 22 2.3 The neighbourhood algorithm . 23 2.4 Conditional parameter spaces . 25 2.5 Conclusions . 28 i ii CONTENTS 3 Forward computation 29 3.1 Dispersion Curves . 29 3.1.1 Propagator-matrix method . 30 3.1.2 Displacements, Stresses, and strains . 31 3.1.3 Eigenvalue problem for Love waves . 31 3.1.4 Eigenvalue problem for Rayleigh waves . 35 3.1.5 A quick root search . 40 3.1.6 Mode jumping control . 44 3.1.7 Misfit . 46 3.1.8 Sensitivity of the dispersion against layer parameters . 46 3.1.9 Conclusion . 56 3.2 Ellipticity . 56 3.2.1 Computation . 57 3.2.2 Sensitivity . 58 3.2.3 Misfit . 60 3.3 Spatial auto-correlation . 61 3.3.1 Computation . 61 3.3.2 Misfit . 62 3.3.3 Sensitivity . 63 3.4 Conclusion . 64 4 Parameterization of a ground model 65 4.1 Theoretical model used in parameterization tests . 65 4.2 Thickness, Vp, and Vs . 67 4.2.1 Two layers . 69 4.2.2 Three layers . 72 4.3 Stack of N layers . 79 4.3.1 Arbitrary profile . 79 4.3.2 Vs inversion without LVZ . 81 4.4 Non-uniform layers . 83 4.4.1 Linear variation . 83 4.4.2 Power law variation . 83 4.5 Conclusions . 86 5 Enhanced inversions 89 5.1 Multimodal curves . 89 5.1.1 Rayleigh higher modes . 90 5.1.2 Love and Rayleigh . 95 5.1.3 Higher mode identification . 95 5.2 Spatial auto-correlation . 100 5.2.1 Uniqueness of auto-correlation curves . 101 CONTENTS iii 5.2.2 Synthetic model . 101 5.2.3 Validation of auto-correlations . 101 5.2.4 Inversion . 104 5.3 Ellipticity inversion . 106 6 Test cases 109 6.1 Synthetic ambient vibrations . 109 6.1.1 Model description . 109 6.1.2 Single source wavefield . 112 6.1.3 Frequency-wavenumber method . 114 6.1.4 High resolution method . 117 6.1.5 Spatial auto-correlation method . 119 6.1.6 Discussion and Conclusions . 120 6.2 Li`ege site . 124 6.2.1 The test site . 124 6.2.2 Ambient vibrations recording . 130 6.2.3 Frequency-wavenumber method . 131 6.2.4 High resolution method . 136 6.2.5 Spatial auto-correlation method . 137 6.2.6 Conclusions . 140 Conclusion 141 A Sub-determinants of R(i) 145 B Generating increasing velocity profiles 149 B.1 Selection method . 150 B.2 Sorting method . 150 B.3 Velocity-jump method . 150 B.4 Interpolation method . 151 B.5 Interpolation method with random start . 151 B.6 Bissection method . 152 B.7 Diagonal method . 152 B.8 Including Poisson's ratio . 153 B.9 Conclusions . 153 References 155 iv CONTENTS List of Figures 1.1 Theoretical array responses for 25 sensors . 8 1.2 Theoretical array responses for 10 sensors . 9 1.3 Reference model for refraction synthetic traveltime-distance plot . 15 1.4 Inversion of synthetic traveltime-distance plot . 16 1.5 Theoretical array responses for a line of sensors . 18 1.6 Example of a f-k analysis for triggered surface waves . 18 2.1 Definition of an inversion problem . 19 2.2 Voronoi cells for a two-dimensional parameter space . 24 2.3 Comparison of variable transformation and selection methods . 26 2.4 High level condition intersection with Voronoi cells . 27 3.1 Schematic one-dimensional model . 30 3.2 Values taken by l21(z0) . 35 3.3 Values taken by R1212(z0) . 39 3.4 Velocity limits of Love and Rayleigh dispersion curves . 40 3.5 Method for bracketing roots. 42 3.6 Method for refining roots . 43 3.7 Dispersion curve of a model with a LVZ. 45 3.8 Two layers: influence of Vs0 with a constant Vp profile . 47 3.9 Two layers: influence of Vs0 with a constant Poisson's ratio . 48 3.10 Two layers: influence of Vp0 . 49 3.11 Two layers: influence.