Physics for Geologists, Second edition

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136 Some dangers of mathematical statistics Table 13.2 Measured values of the solid proportion Vo, of 5 550 m s-' . Linear regression analysis gives the formula with a correlation coefficient of r = 0.90 with 11 degrees of freedom, which is highly significant. This seems to be entirely satisfactory. However, linear regression analysis of the logarithms of the quantities reveals that the natural relationship cannot be linear because the slope is much nearer 0.5 than 1. The correlation coefficient is also 0.90. Linear regression analysis of the logarithms leads to the following equation: Now, one boundary condition is that when the porosity is zero and 1 -f = 1, the dimensionless velocity V/Vo = 1. The coefficient 6 must therefore also equal 1, not 0.97. If we constrain the line so that it passes through the points In (1,l) and the mean values of In (1 - f ) and In (V/Vo) the slope is 0.52 and a good practical formula turns out to be This satisfies the boundary conditions, and it only remains to determine the best value of Vo to use. This is a very different formula from the one that Copyright 2002 by Richard E. Chapman
Some dangers of mathematical statistics 137 describes seismic P waves, probably because the wavelengths of seismic P waves are very much longer than those of an acoustic borehole logging tool. The P wave generated by an earthquake has a frequency of about one hertz and a velocity of about 5 kms-I, so its wavelength is about 5 km. The P wave generated in seismic exploration has a frequency of about 50 Hz and a velocity of about 3 km s-l, so its wavelength is of the order of 60 m. The acoustic log signal (20 kHz) has character over less than a metre and so its wavelength is very much less than a metre - in fact, about 15 cm. Any formula representing geological phenomena must be looked at care- fully. At the end of Chapter 9 we looked at compaction curves, and came to the conclusion that the common formula for the normal compaction trend of mudstones with depth, as measured by the sonic or acoustic log, could not possibly be right because as the dimensionless depth z/b becomes very large, the shale transit time tends to zero, rather than to the transit time when the porosity is zero. This we called the matrix transit time. See Moroney (1956) for a painless but instructive introduction to statistical methods, even if he predated the pocket calculator days. See also Rosen (1992: 178f.) for a good summary of the mathematical statistical techniques. Copyright 2002 by Richard E. Chapman
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Physics for Geologists Second editi
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Copyright 2002 by Richard E. Chapma
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... vlii Contents Polarization 47 L
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Figures Copyright 2002 by Richard E
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Tables Copyright 2002 by Richard E.
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xiv Preface waves diminishes with d
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xvi Preface provides. This book is
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xviii Acknowledgements to K. Whaler
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xx Symbols mass refractive index bu
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2 Basic concepts Table 1.1 Dimensio
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4 Basic concepts Table 1.2 Basic SI
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6 Basic concepts pascal (Pa) and si
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8 Basic concepts all the relevant d
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10 Basic concepts relating them by
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2 Force The dimensions of force are
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14 Force The ratio of the Moon's ac
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16 Force Figure 2.1 The sum of the
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18 Force When a motorcycle takes a
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20 Force Figure 2.3 Torque is the p
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22 Force How does momentum differ f
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24 Force the weights - that is, the
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26 Force Figure 2.6 Wall-sticking i
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3 Gravity Universal gravity When di
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30 Gravity are significant to geolo
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32 Gravity Figure 3.2 The tractive,
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34 Gravity Mean sea level It might
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36 Gravity The benefits of space fl
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38 Gravity Figure 3.6 Profile of th
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40 Gravity Figure 3.8 The siphon at
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4 Optics Light is an electromagneti
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44 Optics Figure 4.2 Refraction. (a
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46 Optics The coeficient of reflect
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48 Optics only the light oscillatin
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50 Optics called Iceland Spar. Two
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52 Optics Diffraction If you shine
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54 Optics Figure 4.7 Geometrical op
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56 Optics Figure 4.9 The mirrors in
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5 Atomic structure and age-dating P
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60 Atomic structure and age-dating
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66 Electromagnetic radiation radiat
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68 Electromagnetic radiation Source
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Heat and heat flow The noun heat ha
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72 Heat and heat flow Second Law: A
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8 Electricity and magnetism Electri
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Electricity and magnetism 77 Figure
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Electricity and magnetism 79 rivett
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Electricity and magnetism 8 1 defin
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Electricity and magnetism 83 is kno
Page 101 and 102: 9 Stress and strain Pressure, p, is
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Page 105 and 106: These three elastic constants are r
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Page 113 and 114: Fracture Stress and strain 97 We fo
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Page 117 and 118: 10 Sea waves The nature of water wa
Page 119 and 120: Sea waves 103 Figure 10.1 Wave moti
Page 121 and 122: Sea waves 105 refraction by islands
Page 123 and 124: Acoustics: sound and other waves 10
Page 125 and 126: Acoustics: sound and other waves 10
Page 127 and 128: 12 Fluids and fluid flow Introducti
Page 129 and 130: Fluids and fluid flow 1 13 Figure 1
Page 131 and 132: Fluids and fluid flow 115 Figure 22
Page 133 and 134: Fluids and fluid flow 117 and the f
Page 135 and 136: water above a vertical thickness h
Page 137 and 138: Fluids and fluid flow 121 where V i
Page 139 and 140: Fluids and fluid flow 123 ardentes,
Page 141 and 142: Fluids and fluid flow 125 Figure 12
Page 143 and 144: - 14 $ 12 E 10 K .- 3 8 r cll + 6 0
Page 145 and 146: Fluids and fluid flow 129 the liqui
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Page 155 and 156: Appendix 139 What would happen if y
Page 157 and 158: Notes 141 and since x is very large
Page 159 and 160: References Allum, J. A. E. (1966) P
Page 161 and 162: References 145 -(1950) 'Mechanism o
Page 163: Further reading 147 Trigg, G. L. an
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Physics for Geologists, Second edition

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