LINRORT - linearized P-P, P-S1 and P-S2 reflection coefficients
for a horizontal interface separating two of any of the
following halfspaces: ISOTROPIC, VTI, HTI and ORTHORHOMBIC.
linrort [optional parameters]
hspace1=ISO medium type of the incidence halfspace:
=ISO ... isotropic
=VTI ... VTI anisotropy
=HTI ... HTI anisotropy
=ORT ... ORTHORHOMBIC anisotropy
for ISO:
vp1=2 P-wave velocity, halfspace1
vs1=1 S-wave velocity, halfspace1
rho1=2.7 density, halfspace1
for VTI:
vp1=2 P-wave vertical velocity (V33), halfspace1
vs1=1 S-wave vertical velocity (V44=V55), halfspace1
rho1=2.7 density, halfspace1
eps1=0 Thomsen's generic epsilon, halfspace1
delta1=0 Thomsen's generic delta, halfspace1
gamma1=0 Thomsen's generic gamma, halfspace1 ",
for HTI:
vp1=2 P-wave vertical velocity (V33), halfspace1
vs1=1 "fast" S-wave vertical velocity (V44), halfspace1
rho1=2.7 density, halfspace1
eps1_v=0 Tsvankin's "vertical" epsilon, halfspace1
delta1_v=0 Tsvankin's "vertical" delta, halfspace1
gamma1_v=0 Tsvankin's "vertical" gamma, halfspace1 ",
for ORT:
vp1=2 P-wave vertical velocity (V33), halfspace1
vs1=1 x2-polarized S-wave vertical velocity (V44), halfspace1
rho1=2.7 density, halfspace1
eps1_1=0 Tsvankin's epsilon in [x2,x3] plane, halfspace1
delta1_1=0 Tsvankin's delta in [x2,x3] plane, halfspace1
gamma1_1=0 Tsvankin's gamma in [x2,x3] plane, halfspace1
eps1_2=0 Tsvankin's epsilon in [x1,x3] plane, halfspace1
delta1_2=0 Tsvankin's delta in [x1,x3] plane, halfspace1
gamma1_2=0 Tsvankin's gamma in [x1,x3] plane, halfspace1
delta1_3=0 Tsvankin's delta in [x1,x2] plane, halfspace1
hspace2=ISO medium type of the reflecting halfspace (the same
convention as above)
medium parameters of the 2nd halfspace follow the same convention
as above:
vp2=2.5 vs2=1.2 rho2=3.0
eps2=0 delta2=0
eps2_v=0 delta2_v=0 gamma2_v=0
eps2_1=0 delta2_1=0 gamma2_1=0
eps2_2=0 delta2_2=0 gamma2_2=0
delta2_3=0
(note you do not need "gamma2" parameter for evaluation
of weak-anisotropy reflection coefficients)
a_file=-1 the string '-1' ... incidence and azimuth angles are
generated automatically using the setup values below
a_file=file_name ... incidence and azimuth angles are
read from a file "file_name"; the program expects a
file of two columns [inc. angle, azimuth]
in the case of a_file=-1:
fangle=0 first incidence phase angle
langle=30 last incidence angle
dangle=1 incidence angle increment
fazim=0 first azimuth (in deg)
lazim=0 last azimuth (in deg)
dazim=1 azimuth increment (in deg)
kappa=0. azimuthal rotation of the lower halfspace2 (e.t. a
symmetry axis plane for HTI, or a symmetry plane for
ORTHORHOMBIC) with respect to the x1-axis
out_inf=info.out information output file
out_P=Rpp.out file with Rpp reflection coefficients
out_S=Rps.out file with Rps reflection coefficients
out_SVSH=Rsvsh.out file with SV and SH projections of reflection
coefficients
out_Error=error.out file containing error estimates evaluated during
the computation of the reflection coefficients;
Output:
out_P:
inc. phase angle, azimuth, reflection coefficient; for a_file=-1, the
inc. angle is the fast dimension
out_S:
inc. phase angle, azimuth, Rps1, Rps2, cos(PHI), sin(PHI); for
a_file=-1, the inc. angle is the fast dimension ",
out_SVSH:
inc. phase angle, azimuth, Rsv, Rsh, cos(PHI), sin(PHI); for
a_file=-1, the inc. angle is the fast dimension
out_Error:
error estimates of Rpp, Rpsv and Rpsh approximations; global error is
analysed as well as partial contributions to the error due to the
isotropic velocity contrasts, and due to anisotropic upper and lower
halfspaces. The error file is self-explanatory, see also descriptions
of subroutines P_err_2nd_order, SV_err_2nd_order and SH_err_2nd_order.
Adopted Convention:
The right-hand Cartesian coordinate system with the x3-axis pointing
upward has been chosen. The upper halfspace (halfspace1)
contains the incident P-wave. Incidence angles can vary from <0,PI/2),
azimuths are unlimited, +azimuth sense counted from x1->x2 axes
(azimuth=0 corresponds to the direction of x1-axis). In the current
version, the coordinate system is attached to the halfspace1 (e.t.
the symmetry axis plane of HTI halfspace1, or one of symmetry planes
of ORTHORHOMBIC halfspace1, is aligned with the x1-axis), however, the
halfspace2 can be arbitrarily rotated along the x3-axis with respect
to the halfspace1. The positive weak-anisotropy polarization of the
reflected P-P wave (e.t. positive P-P reflection coefficient) is close
to the direction of isotropic slowness vector of the wave (pointing
outward the interface). Similarly, weak-anisotropy S-wave reflection
coefficients are described in terms of "SV" and "SH" isotropic
polarizations, "SV" and "SH" being unit vectors in the plane
perpendicular to the isotropic slowness vector. Then, the positive
"SV" polarization vector lies in the incidence plane and points
towards the interface, and positive "SH" polarization vector is
perpendicular to the incidence plane, aligned with the positive
x2-axis, if azimuth=0. Rotation angle "PHI", characterizing a
rotation of "the best projection" of the S1-wave polarization
vector in the isotropic SV-SH plane in the incidence halfspace1, is
counted in the positive sense from "SV" axis (PHI=0) towards the
"SH" axis (PHI=PI/2). Of course, S2 is perpendicular to S1, and
the projection of S1 and S2 polarizations onto the SV-SH plane
coincides with SV and SH directions, respectively, for PHI=0.
The units for velocities are km/s, angles I/O are in degrees
Additional Notes:
The coefficients are computed as functions of phase incidence
angle and azimuth (determined by the incidence slowness vector).
Vertical symmetry planes of the HTI and
ORTHORHOMBIC halfspaces can be arbitrarily rotated along the
x3-axis. The linearization is based on the assumption of weak ",
contrast in elastic medium parameters across the interface,
and the assumption of weak anisotropy in both halfspaces.
See the "Adopted Convention" paragraph below for a proper
input.
Author: Petr Jilek, CSM-CWP, December 1999.