Annual Report 2000 - WIT
Annual Report 2000 - WIT Annual Report 2000 - WIT
160 1982). There is also the so-called traveltime-corrected meanfield formalism which is successfully applied to interpret attenuation measurements in seismology (Sato and Fehler, 1998). The traveltime-corrected meanfield formalism however is based on heuristic assumptions. Stronger wavefield fluctuations visible through a longer and more complex coda indicate that the process of multiple scattering becomes important. Then the radiative transfer theory and the diffusion approximation are adequate concepts to describe averaged wavefields . In addition numerous numerical studies characterized the scattering attenuation in heterogenous structures (e.g. Frankel and Clayton, 1986). Several pulse propagation theories for random media already appeared in the literature. Wenzel (1975) describes the propagation of coherent transients with perturbation theory. Beltzer (1989) investigated the pulse propagation in heterogeneous media based on the causality principle. The above mentioned wavefield approximations describe the wavefield for an ensemble of medium's realizations. This is because ensemble averaged wavefield attributes are used. However, in geophysics only one realization of the heterogenous medium exists. Then the ensemble averaged wavefield attribute does only represent the measured one provided that it is a self-averaged wavefield attribute (Gredeskul and Freilikher, 1990). That is to say it becomes averaged while propagating inside the random medium. A pulse propagation theory valid for single realizations of 2-D and 3-D random media is not available. In the case of 1-D random media such a wavefield description is known as the generalized O'Doherty-Anstey (ODA) formalism (Shapiro and Hubral, 1999). It is a suitable wavefield description of the primary wavefield, i.e., the wavefield in the vicinity of the first arrival. There are also recent efforts to generalize the ODA theory for so-called locally layered media, i.e., media with layered microstructure but a 3-D heterogeneous macroscopic background (Solna and Papanicolaou, 2000). The purpose of this study is to present a method that allows to describe pulse propagation in single realizations of 2-D and 3-D random media. It is based on ensembleaveraged, logarithmic wavefield attributes derived in the Rytov approximation which is only valid in the weak wavefield fluctuation regime. Müller and Shapiro (2000) numerically showed the partial self-averaging of logarithmic wavefield attributes and constructed the medium's Green function for an incident plane wave. Here, we follow the same strategy, however using logarithmic wavefield attributes that additionally fulfill the Kramers-Kronig relations and are valid in a very broad frequency range.
Ç Ò ¥ Ì äÒ Ö Òh–šŠˆ?•“˜0Ÿ Ö Ò ˆ { × k Ç Ò 161 THEORY Rytov transformation and the generalized O'Doherty-Anstey theory The basic mathematical model for scalar wave propagation in heterogeneous media is the wave equation with random coefficients. In the following, we look for a solution of the acoustic wave equation ‘uÎ (1) â—–šŠˆ?•˜)Ÿ¬ šj˜)Ÿ Ò –šj˜$•.ˆ=Ÿ with as a scalar wavefield (in the following simply denoted – by ). Here we defined the squared slowness as šj˜)Ÿ ‘ šœÌA¹“¤ šj˜)Ÿ=Ÿ • (2) Ò where ä denotes the propagation velocity in a homogeneous reference medium. The ¥ šš˜)Ÿ function is a realization of a stationary statistically isotropic random field with zero average, ›Wšj˜)Ÿ“œ‘ i.e., Ÿ!Wšj˜ Ÿ“œ that only depends on the difference coordinate ë ‘ ó ˜ ¬ž˜ šŠëŸP‘~›Wšj˜ Later we make use the exponential correlation šŠëŸÏ‘ Ÿ'Ò åjæÄçèšœ¬ó ëó])êÄŸ function , Ÿ Ò where •«ê denote the variance of the fluctuations and correlation length, respectively. k and is characterized by a spatial correlation function Î . ó We consider now a time-harmonic plane wave propagating in a 2-D and 3-D random medium and describe the wavefield inside the random medium with help of the Rytov transformation, using the complex exponent ‘S¡üÁºN® , where the real part ¡ represents the fluctuations of the logarithm of the amplitude and the imaginary part ® represents phase fluctuations. Omitting the time dependence åjæÄçèšF¬´ºŠétˆ=Ÿ , we write ‘ –䦚 镘)Ÿ–åjæÄ禚š¡Ašé•˜)Ÿ'¯º7®_š 镘)Ÿ.Ÿ°• (3) –šé•˜¦Ÿ –ä ‘ ¢läåjæÄç蚊º7® äjŸ where is the wavefield in the homogeneous reference medium ‘ Î ) with the amplitude ¢´ä and the unwrapped phase ® ä . To be more specific, we assume that the initially plane wave propagates vertically along the (Wšj˜)Ÿ z-axis. Equation (7) can then be written where we introduced the complex wavenumber –šé•ÆŽá‘S£ •˜ ¤°ŸK‘ì í@¥ ¦@¥ § ¨@"#¦ } • (4) (5) é•h£ •˜ ¤'Ÿ ®_š £ ®ä¦šé•£AŸ ± £ •˜ ¤¦Ÿ ¡Ašé•h£ £ š é•h£ •˜ ¤'Ÿ°• (6) ¬ º À¯š é•h£ •˜ ¤'Ÿ ‘ £ ‘ © š é•h£ •˜ ¤'Ÿ¦Áº
- Page 122 and 123: 110 et al., 1992), Hanitzsch (1997)
- Page 124 and 125: 112 consecutive wavefronts). Howeve
- Page 126 and 127: 114 Numerical example We test the W
- Page 128 and 129: 116 Versteeg, R., and Grau, G., 199
- Page 130 and 131: 118 indicator. To extract elastic p
- Page 132 and 133: € b b 120 S where is the position
- Page 134 and 135: É É É É 122 The À constant (da
- Page 136 and 137: 124 0 Distance [ m ] 1000 2000 3000
- Page 138 and 139: 126 Kosloff, D., Sherwood, J., Kore
- Page 140 and 141: 128
- Page 142 and 143: 130 penetration distances and a pot
- Page 144 and 145: and » ˆ Ö Ö is the scalar hydra
- Page 146 and 147: Å éÙó ó ».-.‹œì/-j쨋
- Page 148 and 149: 136 TRIGGERING FRONTS IN HETEROGENE
- Page 150 and 151: Ö ˆ Ö “P£'ˆ é ˆ ó,Lˆ¨
- Page 152 and 153: Ö Ö Ê » Ø Ö » Ö Ê Ê Ê Ö
- Page 154 and 155: 142 300 250 200 a) eikonal equation
- Page 156 and 157: 144 Shapiro, S. A., and Müller, T.
- Page 158 and 159: » i » m ×]Ú ‘Œƒšj'Ÿlk hM
- Page 160 and 161: 148 RESULTS Soultz-sous-Forêts Inv
- Page 162 and 163: …„ƒ 150 for the smaller ellips
- Page 164 and 165: 152 Figure 5: The cloud of events f
- Page 166 and 167: …„ƒ Î Î ÎÄÈ‹•¨ Î-È
- Page 168 and 169: 156 straightforward. The new approa
- Page 170 and 171: 158
- Page 174 and 175: { ³ u ´ ´ ´ ´ -y ›-^œ ´ ´
- Page 176 and 177: Ï u u ¬ 164 To summarize, we deri
- Page 178 and 179: Ý ì á ä é å¤æDçzè ä é
- Page 180 and 181: ø M ã = ã Ù ø ä Ú ã Ù ø
- Page 182 and 183: ã Q c ã ä 170 a=10m; std.dev.=8%
- Page 184 and 185: 172 Frankel, A., and Clayton, R. W.
- Page 186 and 187: 174 It is well-known that inhomogen
- Page 188 and 189: ã ä ä ŸSŸ S ¡S¡ ©©¨¨ æ
- Page 190 and 191: Kneib, 1995 285-6000 superposition
- Page 192 and 193: 180 parameter: Hurst coefficient pa
- Page 194 and 195: 182 REFERENCES Bourbié, T., Coussy
- Page 196 and 197: 184 1999) in multiple fractured med
- Page 198 and 199: ‹œž Ÿ¢¡¤£¦¥ƒ§©¨ ‘
- Page 200 and 201: j jÆÅÇ[È”u j jÎÅÇ[È”uw
- Page 202 and 203: 190 Davis, P. M., and Knopoff, L.,
- Page 204 and 205: 192 properties from the measured se
- Page 206 and 207: 194 discussion of these problems ca
- Page 208 and 209: 196 Altogether, close to 260 shotpo
- Page 210 and 211: 198 Time (ms) 0 2 4 6 8 10 12 14 16
- Page 212 and 213: 200 of groundwater. ACKNOWLEDGEMENT
- Page 214 and 215: 202
- Page 216 and 217: 204 the wavefront curvature matrix
- Page 218 and 219: û ò é from ã äñ to ã î§ñ
- Page 220 and 221: é 208 Table 1: Median relative err
Ç<br />
Ò<br />
¥<br />
Ì<br />
äÒ<br />
Ö<br />
Òh–šŠˆ?•“˜0Ÿ<br />
Ö<br />
Ò ˆ<br />
{<br />
×<br />
k<br />
Ç<br />
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161<br />
THEORY<br />
Rytov transformation and the generalized O'Doherty-Anstey theory<br />
The basic mathematical model for scalar wave propagation in heterogeneous media is<br />
the wave equation with random coefficients. In the following, we look for a solution<br />
of the acoustic wave equation<br />
‘uÎ (1)<br />
â—–šŠˆ?•˜)Ÿ¬<br />
šj˜)Ÿ Ò<br />
–šj˜$•.ˆ=Ÿ with as a scalar wavefield (in the following simply denoted – by ). Here we<br />
defined the squared slowness as<br />
šj˜)Ÿ ‘ šœÌA¹“¤ šj˜)Ÿ=Ÿ • (2)<br />
Ò<br />
where ä denotes the propagation velocity in a homogeneous reference medium. The<br />
¥<br />
šš˜)Ÿ function is a realization of a stationary statistically isotropic random field with<br />
zero average, ›Wšj˜)Ÿ“œ‘ i.e.,<br />
Ÿ!Wšj˜ Ÿ“œ that only depends on the difference coordinate ë ‘ ó ˜ ¬ž˜<br />
šŠëŸP‘~›Wšj˜<br />
Later we make use the exponential correlation šŠëŸÏ‘ Ÿ'Ò åjæÄçèšœ¬ó ëó])êÄŸ function ,<br />
Ÿ Ò where •«ê denote the variance of the fluctuations and correlation length, respectively.<br />
k<br />
and is characterized by a spatial correlation function<br />
Î<br />
. ó<br />
We consider now a time-harmonic plane wave propagating in a 2-D and 3-D random<br />
medium and describe the wavefield inside the random medium with help of the<br />
Rytov transformation, using the complex exponent ‘S¡üÁºN® , where the real part ¡<br />
represents the fluctuations of the logarithm of the amplitude and the imaginary part ®<br />
represents phase fluctuations. Omitting the time dependence åjæÄçèšF¬´ºŠétˆ=Ÿ , we write<br />
‘ –䦚 镘)Ÿ–åjæÄ禚š¡Ašé•˜)Ÿ'¯º7®_š 镘)Ÿ.Ÿ°• (3)<br />
–šé•˜¦Ÿ<br />
–ä ‘ ¢läåjæÄç蚊º7® äjŸ where is the wavefield in the homogeneous reference medium<br />
‘ Î ) with the amplitude ¢´ä and the unwrapped phase ® ä . To be more specific,<br />
we assume that the initially plane wave propagates vertically along the<br />
(Wšj˜)Ÿ<br />
z-axis.<br />
Equation (7) can then be written<br />
where we introduced the complex wavenumber<br />
–šé•ÆŽá‘S£ •˜ ¤°ŸK‘ì<br />
<br />
í@¥ ¦@¥ § ¨@"#¦<br />
}<br />
• (4)<br />
(5)<br />
é•h£ •˜ ¤'Ÿ ®_š<br />
£<br />
®ä¦šé•£AŸ<br />
± £<br />
•˜ ¤¦Ÿ ¡Ašé•h£<br />
£<br />
š é•h£ •˜ ¤'Ÿ°• (6)<br />
¬ º<br />
À¯š é•h£ •˜ ¤'Ÿ ‘ £<br />
‘ © š é•h£ •˜ ¤'Ÿ¦Áº
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