Prevailing-frequency approximation of the coupling ray theory for S waves
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1 Prevailing-frequency approximation of the coupling ray theory for S waves Petr Bulant & Luděk Klimeš Department of Geophysics Faculty of Mathematics and Physics Charles University in Prague S EI S MIC WA VE S I N C OM PLEX 3 D S U RE U CT S TR
2 Isotropic ray theory Applicable to isotropic and very weakly anisotropic media. Anisotropic ray theory Always applicable to P waves. Applicable to S waves in strongly anisotropic media. Coupling ray theory (Coates & Chapman, 1990) Frequency dependent S wave polarization. Applicable to isotropy and to all degrees of anisotropy. Low frequency limit: Isotropic ray theory. High frequency limit: Anisotropic ray theory. Usually calculated along the common S wave reference ray.
3 S-wave polarization along a ray: Isotropic ray theory polarization (ω = 0) Coupling ray theory polarization (0 < ω < ) Anisotropic ray theory polarization (ω = )
4 Comparison of isotropic, coupling and anisotropic ray theories in velocity model QIH VERTICAL FORCE EARTH SURFACE RECEIVER DEPTH / km SOURCE-VSP HORIZONTAL DISTANCE = 1.00 km
5 X3 Coupling ray theory in model QIH (transverse component) 0.5 TIME X3 Anisotropic ray theory in model QIH (transverse component) 0.5 TIME
6 Development of S-wave splitting with increasing anisotropy in velocity models QIH, QI, QI2 and QI4 (anisotropy ratio 1 : 2 : 4 : 8) VERTICAL FORCE EARTH SURFACE RECEIVER DEPTH / km SOURCE-VSP HORIZONTAL DISTANCE = 1.00 km
7 Coupling-ray-theory seismograms in models QIH and QI (transverse component) TIME X3 TIME X3
8 Coupling-ray-theory seismograms in models QI and QI2 (transverse component) TIME X3 TIME X3
9 Coupling-ray-theory seismograms in models QI2 and QI4 (transverse component) TIME X3 TIME X3
10 Idea of the prevailing-frequency approximation Anisotropic-ray-theory Green tensor: 2 G ART ij (x,x 0,ω) = A (K) (x,x 0 ) g (K) i (x)g (K) j (x 0 ) exp[iωτ (K) (x,x 0 )] K=1 Coupling-ray-theory Green tensor: G CRT ij (x,x 0,ω) Prevailing-frequency approximation of the coupling-ray-theory Green tensor: G CRT ij (x,x 0,ω) G PFA ij (x,x 0,ω) Prevailing-frequency Green tensor: 2 G PFA ij (x,x 0,ω)= A (K) (x,x 0 )e (K) (x,ω 0 )e (K) j (x 0,ω 0 )exp [ iωt (K) (x,x 0,ω 0 ) ] K=1 ω 0... prevailing circular frequency. i T (K) (x,x 0,ω 0 )... coupling-ray-theory travel times. (x,ω 0 ), e (K) (x 0,ω 0 )... coupling-ray-theory polarization vectors. e (K) i j
11 Prevailing-frequency Green tensor conditions: G PFA ij (x,x 0,ω 0 ) = G CRT ij (x,x 0,ω 0 ) G PFA ij (x,x 0,ω 0 ) = GCRT ij ω (x,x 0,ω 0 ) ω These conditions uniquely determine coupling-ray-theory travel times T (K) (x,x 0,ω 0 ) and coupling-ray-theory polarization vectors e (K) i (x,ω 0 ) and e (K) j (x 0,ω 0 ). We numerically calculate G CRT ij (x,x 0,ω 0 ) using the algorithm by Bulant & Klimeš (2002). We calculate GCRT ij ω (x,x 0,ω 0 ) using the derivative of this algorithm.
12 Numerical comparison of seismograms Prevailing-frequency approximation. Standard coupling ray theory. Fourier pseudospectral method. Source-receiver configuration: VERTICAL FORCE EARTH SURFACE RECEIVERS SOURCE-VSP HORIZONTAL DISTANCE = 1.00 km
13 Velocity model QIH, transverse (top) and vertical (bottom) component. Prevailing-frequency approximation, standard coupling ray theory.
14 Velocity model QI, transverse (top) and vertical (bottom) component. Prevailing-frequency, coupling ray theory, Fourier pseudospectral method.
15 Velocity model QI2, transverse (top) and vertical (bottom) component. Prevailing-frequency, coupling ray theory, Fourier pseudospectral method.
16 Velocity model QI4, transverse (top) and vertical (bottom) component. Prevailing-frequency, coupling ray theory, Fourier pseudospectral method.
17 Velocity model KISS, transverse (top) and vertical (bottom) component. Prevailing-frequency, coupling ray theory, Fourier pseudospectral method.
18 Velocity model SC1 I, transverse (top) and vertical (bottom) component. Prevailing-frequency, coupling ray theory, Fourier pseudospectral method.
19 Velocity model SC1 II, transverse (top) & vertical (bottom) component. Prevailing-frequency, coupling ray theory, Fourier pseudospectral method.
20 Velocity model ORT, transverse (top) and vertical (bottom) component. Prevailing-frequency, coupling ray theory, Fourier pseudospectral method.
21 Velocity model QI2, transverse (top) and vertical (bottom) component. Broad band 3 Hz 80 Hz. Prevailing-frequency, coupling ray theory.
22 Reference rays for the coupling ray theory Isotropic common reference rays Traced in the isotropic reference velocity model. Least accurate. Anisotropic common reference rays Traced using the averaged Hamiltonian function of both anisotropic-raytheory S waves. Universal option. Always better than the isotropic common reference rays. Anisotropic-ray-theory rays Usually unusable. Brief demonstration will follow. SH and SV reference rays Very accurate in generally anisotropic velocity models with considerable transversely isotropic components. Demonstration of accuracy will follow.
23 Anisotropic-ray-theory rays of the faster S wave in velocity model SC1 II.
24 Anisotropic-ray-theory rays of the slower S wave in velocity model SC1 II.
25 Ray-velocity surface of the faster S wave in a triclinic elastic medium.
26 Ray-velocity surface of the slower S wave in a triclinic elastic medium.
27 Detail of the S-wave ray-velocity surfaces in a triclinic elastic medium.
28 SH and SV reference rays In generally anisotropic velocity models with considerable transversely isotropic components (Klimeš, 2015), we trace the SH and SV reference rays according to Klimeš & Bulant (2015). We apply the prevailing-frequency approximation of the coupling ray theory to the SH and SV reference rays. We obtain two S-wave arrivals along each SH ray and have to select the right one of them. Analogously, we obtain two S-wave arrivals along each SV ray and have to select the right one of them. We select the right arrivals according to their polarization and travel time (Klimeš & Bulant, 2014).
29 Accuracy of the synthetic seismograms calculated along the SH and SV reference rays The red seismograms are calculated using the prevailing-frequency approximation of the coupling ray theory (a) along the anisotropic common S-wave rays with the quadratic perturbation expansions of travel times, and (b) along the SH and SV reference rays with the linear perturbation expansions of travel times. They are overlaid by the black seismograms calculated using the Fourier pseudospectral method by Pšenčík, Farra & Tessmer (2012) which is considered here as a nearly exact reference.
30 Model QI2, vertical component Anisotropic common S-wave reference rays.
31 Model QI2, vertical component SH and SV reference rays.
32 Model QI2, radial component Anisotropic common S-wave reference rays.
33 Model QI2, radial component SH and SV reference rays.
34 Model QI2, transverse component Anisotropic common S-wave reference rays.
35 Model QI2, transverse component SH and SV reference rays.
36 Model QI4, vertical component Anisotropic common S-wave reference rays.
37 Model QI4, vertical component SH and SV reference rays.
38 Model QI4, radial component Anisotropic common S-wave reference rays.
39 Model QI4, radial component SH and SV reference rays.
40 Model QI4, transverse component Anisotropic common S-wave reference rays.
41 Model QI4, transverse component SH and SV reference rays.
42 Model SC1 I, vertical component Anisotropic common S-wave reference rays.
43 Model SC1 I, vertical component SH and SV reference rays.
44 Model SC1 I, radial component Anisotropic common S-wave reference rays.
45 Model SC1 I, radial component SH and SV reference rays.
46 Model SC1 I, transverse component Anisotropic common S-wave reference rays.
47 Model SC1 I, transverse component SH and SV reference rays.
48 Model SC1 II, vertical component Anisotropic common S-wave reference rays.
49 Model SC1 II, vertical component SH and SV reference rays.
50 Model SC1 II, radial component Anisotropic common S-wave reference rays.
51 Model SC1 II, radial component SH and SV reference rays.
52 Model SC1 II, transverse component Anisotropic common S-wave reference rays.
53 Model SC1 II, transverse component SH and SV reference rays.
54 Interpolation of the ray-theory Green tensor within ray cells Interpolation of the ray-theory Green tensor within ray cells using the algorithm designed by Bulant & Klimeš (1999) has been proved especially efficient for calculating the ray-theory Green tensor at the nodes of dense 3-D grids. The discretized ray-theory Green tensor can be used for various applications including the ray-based Born approximation, non-linear determination of seismic hypocentres, studies of seismic sources, or Kirchhoff prestack depth migration. The prevailing-frequency approximation enables us to interpolate the coupling-ray-theory S-wave Green tensor within ray cells using the algorithm designed by Bulant & Klimeš (1999).
55 Continuity of the coupling-ray-theory S-wave Green tensor along rays Both prevailing-frequency coupling-ray-theory Green tensors are calculated along a single anisotropic common S-wave reference ray. The anisotropic common reference rays then determine the ray cells for interpolation. At each point of each reference ray, we have two prevailing-frequency coupling-ray-theory Green tensors. We double each reference ray and match the pair of the prevailingfrequency coupling-ray-theory Green tensors with the pair of new rays so that each Green tensor is continuous along the corresponding new ray.
56 Continuity of the coupling-ray-theory S-wave Green tensor within ray tubes In place of each reference ray, we have two new rays corresponding to two prevailing-frequency coupling-ray-theory Green tensors. The Green tensor is continuous along the corresponding ray. As the result, each of three edges of each old ray tube is represented by two rays instead of one ray. We need to double each old ray tube and match the three pairs of edge rays with the pair of new ray tubes so that the Green tensor is continuous within either of the two new ray tubes. The currently designed algorithm is obviously not optimal. The study of continuity of the prevailing-frequency coupling-ray-theory Green tensor within ray tubes represents the most challenging task.
57 Numerical examples of the interpolation within ray cells
58 Relative coupling-ray-theory S-wave travel-time difference D/τ in velocity model QIH. Colour scale: 0.00%, 0.175%, 0.350%, 0.525%, 0.700%, 0.875%.
59 Relative coupling-ray-theory S-wave travel-time difference D/τ in velocity model QI. Colour scale: 0.00%, 0.35%, 0.70%, 1.05%, 1.40%, 1.75%.
60 Relative coupling-ray-theory S-wave travel-time difference D/τ in velocity model QI2. Colour scale: 0.00%, 0.7%, 1.4%, 2.1%, 2.8%, 3.5%.
61 Relative coupling-ray-theory S-wave travel-time difference D/τ in velocity model QI4. Colour scale: 0.00%, 1.4%, 2.8%, 4.2%, 5.6%, 7.0%.
62 Relative coupling-ray-theory S-wave travel-time difference D/τ in velocity model KISS. Colour scale: 0.00%, 0.35%, 0.70%, 1.05%, 1.40%, 1.75%.
63 Relative coupling-ray-theory S- wave travel-time difference D/τ in velocity model SC1 I. Colour scale: 0.00%, 1.1%, 2.2%, 3.3%, 4.4%, 5.5%.
64 Relative coupling-ray-theory S- wave travel-time difference D/τ in velocity model SC1 II. The diagonal shadow zone is probably caused by too narrow horizontal extent of the angular domain of the initial slowness vector. We have not been able to calculate the coupling-ray-theory S-wave Green tensor outside this narrow angular domain. Colour scale: 0.00%, 0.6%, 1.2%, 1.8%, 2.4%, 3.0%.
65 Conclusions The prevailing-frequency approximation of the coupling ray theory allows us to process the coupling-ray-theory wave field in the same way as the anisotropic-ray-theory wave field. The prevailing-frequency approximation of the coupling ray theory can thus be included in wavefront tracing and in the interpolation within ray cells in anisotropic media. This will enable common-source Kirchhoff prestack depth migration with coupling-ray-theory S waves. In generally anisotropic velocity models with considerable transversely isotropic components, the SH and SV reference rays represent very accurate reference rays for the coupling ray theory. Otherwise, we should use the anisotropic common reference rays.
66 References: Bulant, P. & Klimeš, L. (1999): Interpolation of ray theory traveltimes within ray cells. Geophys. J. int., 139, Bulant, P. & Klimeš, L. (2002): Numerical algorithm of the coupling ray theory in weakly anisotropic media. Pure appl. Geophys., 159, Coates, R.T. & Chapman, C.H. (1990): Quasi-shear wave coupling in weakly anisotropic 3-D media. Geophys. J. int., 103, Klimeš, L. (2015): Determination of the reference symmetry axis of a generally anisotropic medium which is approximately transversely isotropic. Seismic Waves in Complex 3-D Structures, 25, Klimeš, L. & Bulant, P. (2012): Single-frequency approximation of the coupling ray theory. Seismic Waves in Complex 3-D Structures, 22, Klimeš, L. & Bulant, P. (2013): Interpolation of the coupling-ray-theory S-wave Green tensor within ray cells. Seismic Waves in Complex 3-D Structures, 23,
67 Klimeš, L. & Bulant, P. (2014): Prevailing-frequency approximation of the coupling ray theory for S waves along the SH and SV reference rays in a transversely isotropic medium. Seismic Waves in Complex 3-D Structures, 24, Klimeš, L. & Bulant, P. (2014): Anisotropic-ray-theory rays in velocity model SC1 II with a split intersection singularity. Seismic Waves in Complex 3-D Structures, 24, Klimeš, L. & Bulant, P. (2015): Ray tracing and geodesic deviation of the SH and SV reference rays in a heterogeneous generally anisotropic medium which is approximately transversely isotropic. Seismic Waves in Complex 3-D Structures, 25, Pšenčík, I., Farra, V. & Tessmer, E. (2012): Comparison of the FORT approximation of the coupling ray theory with the Fourier pseudospectral method. Stud. geophys. geod., 56,
68 Acknowledgements The research has been supported: by the Grant Agency of the Czech Republic under contract P210/ 10/0736, by the Ministry of Education of the Czech Republic within research projects MSM and CzechGeo/EPOS LM , and by the consortium Seismic Waves in Complex 3-D Structures S EI S MIC WA VE S I N C OM PLEX 3 D S U RE U CT S TR
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