Comparison of the anisotropic-common-ray
approximation of the coupling ray theory for
S waves
with the Fourier pseudo-spectral method
in weakly anisotropic models
Petr Bulant1 , Ivan Pšenčı́k2 , Véronique Farra3 , Ekkehart Tessmer4
1
Department of Geophysics, Faculty of Mathematics and Physics, Charles
University in Prague, Czech Republic, http://sw3d.cz/staff/bulant.htm
2
Institute of Geophysics, Academy of Sciences of the Czech Republic,
http://sw3d.cz/staff/psencik.htm
3
Institut de Physique du Globe de Paris, France
4
Institute of Geophysics, Hamburg, Germany
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12th International Congress of the Brazilian Geophysical Society, Rio de Janeiro, August 15-18, 2011
Introduction
In this paper we investigate S-wave propagation in inhomogeneous weakly
anisotropic media in the vicinity of shear-wave singularities. We compare
synthetic seismograms calculated by different coupling ray theory algorithms with seismograms calculated by Fourier pseudo-spectral method.
Fourier pseudo-spectral method
- applicable to any type and strength of anisotropy
- high numerical costs (≈ hours)
- very accurate method, used here as a reference
Coupling ray theory
- applicable to weakly anisotropic models
- very fast method (≈ tens of seconds)
- recently developed algorithms, accuracy needs to be tested
Coupling ray theory
Traditionally, there are two standard ray theories:
isotropic ray theory
- isotropic medium
- just one S-wave
anisotropic ray theory
- two fully separated S-waves
- good results for stronger anisotropy
- problems for weak anisotropy and for singularity regions
Shear waves propagating in inhomogeneous, weakly anisotropic media in
regions, in which the two S waves propagate with identical or close phase
velocities, do not propagate independently. They are coupled. This effect
is described by coupling ray theory
- generalization of both zero-order anisotropic and isotropic ray theories
- provides continuous transition between them
Common-ray approximation of the coupling ray theory
Coupling ray theory
- amplitudes of the two S waves are computed by solving two coupled,
frequency-dependent differential equations along a reference ray
- theoretically, the best reference ray for each of the two S waves is the
corresponding anisotropic ray
Common-ray approximation of the coupling ray theory
- only one reference ray is traced for both anisotropic-ray-theory S waves,
and both S-wave anisotropic ray-theory travel times in the coupling equations are approximated by the perturbation expansion from the common
reference ray
- eliminates problems with ray tracing through S-wave singularities
- considerably simplifies coding of the coupling ray theory and numerical
calculations
- may introduce errors in wavefield due to the perturbation
Numerical comparissons of coupling ray theory with FM
Configuration
0 . 55
0 . 50
TIME
RECEIVERS
0 . 45
EARTH SURFACE
0 . 40
- 0 . 03
VERTICAL
FORCE
0 . 06
0 . 15
0 . 24
0 . 33
0 . 42
0 . 51
X3
0 . 60
SOURCE-VSP HORIZONTAL DISTANCE = 1.00 km
Methods
Green: coupling ray theory by Bulant and Klimeš (2002) with the first
order perturbations of travel time (CRT)
Red: coupling ray theory algorithm by Farra and Pšenčı́k (2011) based on
first-order rays (FOCRT)
Blue: coupling ray theory by Bulant and Klimeš (2002) with the second
order perturbations of travel time (CRTS)
Black: Fourier pseudo-spectral method (FM)
Models
QI, QI4, SC1, SC2, KISS and ORT
seismograms
QI
differences from FM
radial component
0 . 55
0 . 55
TIME
TIME
0 . 50
0 . 50
0 . 45
0 . 45
0 . 40
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0 . 40
0.0
0.1
0.2
0.3
0.4
0.5
X3
transverse component
0.6
X3
0 . 55
0 . 55
TIME
TIME
0 . 50
0 . 50
0 . 45
0 . 45
0 . 40
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0 . 40
0.0
0.1
0.2
0.3
0.4
0.5
X3
vertical component
0.6
X3
0 . 55
0 . 55
TIME
TIME
0 . 50
0 . 50
0 . 45
0 . 45
0 . 40
0.0
0.1
0.2
0.3
0.4
0.5
0.6
X3
FM, CRT, FOCRT, CRTS
0 . 40
0.0
0.1
0.2
0.3
0.4
0.5
0.6
X3
seismograms
QI4
differences from FM
radial component
0 . 55
0 . 55
TIME
TIME
0 . 50
0 . 50
0 . 45
0 . 45
0 . 40
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0 . 40
0.0
0.1
0.2
0.3
0.4
0.5
X3
transverse component
0.6
X3
0 . 55
0 . 55
TIME
TIME
0 . 50
0 . 50
0 . 45
0 . 45
0 . 40
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0 . 40
0.0
0.1
0.2
0.3
0.4
0.5
X3
vertical component
0.6
X3
0 . 55
0 . 55
TIME
TIME
0 . 50
0 . 50
0 . 45
0 . 45
0 . 40
0.0
0.1
0.2
0.3
0.4
0.5
0.6
X3
FM, CRT, FOCRT, CRTS
0 . 40
0.0
0.1
0.2
0.3
0.4
0.5
0.6
X3
seismograms
SC1
differences from FM
radial component
0.7
0.7
TIME
TIME
0.6
0.6
0.5
0.5
0.4
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
0.4
0.0
0.2
0.4
0.6
0.8
1.0
1.2
X3
transverse component
1.4
X3
0.7
0.7
TIME
TIME
0.6
0.6
0.5
0.5
0.4
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
0.4
0.0
0.2
0.4
0.6
0.8
1.0
1.2
X3
vertical component
1.4
X3
0.7
0.7
TIME
TIME
0.6
0.6
0.5
0.5
0.4
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
X3
FM, CRT, FOCRT, CRTS
0.4
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
X3
seismograms
SC2
differences from FM
radial component
0.7
0.7
TIME
TIME
0.6
0.6
0.5
0.5
0.4
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
0.4
0.0
0.2
0.4
0.6
0.8
1.0
1.2
X3
transverse component
1.4
X3
0.7
0.7
TIME
TIME
0.6
0.6
0.5
0.5
0.4
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
0.4
0.0
0.2
0.4
0.6
0.8
1.0
1.2
X3
vertical component
1.4
X3
0.7
0.7
TIME
TIME
0.6
0.6
0.5
0.5
0.4
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
X3
FM, CRT, FOCRT, CRTS
0.4
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
X3
seismograms
KISS
differences from FM
radial component
1.2
1.2
TIME
TIME
1.1
1.1
1.0
1.0
0.9
0.9
0.8
0.8
0.7
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
0.7
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
X3
transverse component
1.6
X3
1.2
1.2
TIME
TIME
1.1
1.1
1.0
1.0
0.9
0.9
0.8
0.8
0.7
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
0.7
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
X3
vertical component
1.6
X3
1.2
1.2
TIME
TIME
1.1
1.1
1.0
1.0
0.9
0.9
0.8
0.8
0.7
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
X3
FM, CRT, FOCRT, CRTS
0.7
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
X3
seismograms
ORT
differences from FM
radial component
0 . 55
0 . 55
TIME
TIME
0 . 50
0 . 50
0 . 45
0 . 45
0 . 40
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0 . 40
0.0
0.1
0.2
0.3
0.4
0.5
X3
transverse component
0.6
X3
0 . 55
0 . 55
TIME
TIME
0 . 50
0 . 50
0 . 45
0 . 45
0 . 40
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0 . 40
0.0
0.1
0.2
0.3
0.4
0.5
X3
vertical component
0.6
X3
0 . 55
0 . 55
TIME
TIME
0 . 50
0 . 50
0 . 45
0 . 45
0 . 40
0.0
0.1
0.2
0.3
0.4
0.5
0.6
X3
FM, CRT, FOCRT, CRTS
0 . 40
0.0
0.1
0.2
0.3
0.4
0.5
0.6
X3
Conclusions
The standard anisotropic ray theory does not work properly or even fails
when applied to S-wave propagation in inhomogeneous weakly anisotropic
media or in the vicinity of shear-wave singularities, where the two shear
waves propagate with similar phase velocities. The tests described in this
paper clearly show that the coupling ray theory yields results very close
to those generated by the FM, which we consider as a very accurate reference. We showed very good results of coupling ray theory for very weak
anisotropy and even for moderate anisotropy ≈ 10%. We also showed very
good results in the vicinity of intersection singularity, kiss singularity, and
conical singularity.
Acknowledgements
- Luděk Klimeš for his kind guidance of the presented work
- Einar Iversen for motivating discussions on the theory of common ray
approximation
The research has been supported by the Grant Agency of the Czech Republic under Contracts P210/10/0736 and P210/11/0117, by the Ministry of
Education of the Czech Republic within research project MSM0021620860,
by the European Community’s FP7 Consortium Project AIM ’Advanced
Industrial Microseismic Monitoring’, Grant Agreement No. 230669, and by
the members of the consortium “Seismic Waves in Complex 3-D Structures”
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(see “http://sw3d.cz”).