We present a suite of programs that implement decades-old
algorithms for computation of seismic surface wave reflection and
transmission coefficients at a welded contact between two laterally
homogeneous quarter-spaces. For Love as well as Rayleigh waves, the
algorithms are shown to be capable of modelling multiple mode conversions at
a lateral discontinuity, which was not shown in the original publications or
in the subsequent literature. Only normal incidence at a lateral boundary is
considered so there is no Love–Rayleigh coupling, but incidence of any mode
and coupling to any (other) mode can be handled. The code is written in
Python and makes use of SciPy's Simpson's rule integrator and NumPy's linear
algebra solver for its core functionality. Transmission-side results from
this code are found to be in good agreement with those from finite-difference
simulations. In today's research environment of extensive computing power,
the coded algorithms are arguably redundant but SWRT can be used as a
valuable testing tool for the ever evolving numerical solvers of seismic wave
propagation. SWRT is available via GitHub
(
There is a vast body of seismological literature on analytical methods for
describing surface wave propagation in laterally inhomogeneous media where
the lateral heterogeneity takes simple forms such as inclusions of regular
shape or plane interfaces. In particular, models with plane vertical
interfaces where the elastic parameters have a first-order discontinuity
received considerable attention in the 1960s and 1970s as idealizations of
sharp, large-scale heterogeneities that actually exist in the Earth such as
continental margins and grabens. The convenience of a vertical boundary is
that the wavefield can be expressed in terms of incident, reflected and
transmitted waves – this not only yields physical insight into the problem
but also reduces it to the evaluation of reflection and transmission
coefficients at the lateral boundary. In general the reflection and
transmission of surface waves at a plane vertical interface cannot be solved
for exactly because the boundary conditions for stresses and displacements
cannot be satisfied exactly by a field of surface waves alone, which in turn
is because the surface wave eigenfunctions for a (layered) half-space do not
form a complete set of basis functions for the elastic wave equation. The
complete basis set consists of a discrete number of surface wave modes plus a
continuum of homogeneous body waves
We focus on the body wave or ray-theoretical method of
Since SWRT is based on previously published techniques, we only discuss the
implementation of those techniques in the SWRT package. Throughout this
section, we use
On the issue of normalization, the main difference between the SWRT
implementation and the original literature is in the normalization of
eigenfunctions used. The original publications used eigenfunctions
As the Herrera normalization and the consequent relationship between
For the types of problems considered in this study, with the lateral
discontinuity in the
Schematic representation of “Model L”. Layer thicknesses are shown
in km and units for velocity and density are km s
Of the three methods included in this study, this is the one where mode conversions have actually been demonstrated in the published literature. Nonetheless, the method, hereafter referred to simply as the Alsop method, is included here for its remarkable simplicity.
In this study
Transmission surface ratio obtained by the GAl method (solid lines)
and GF method (dashed lines with circles), for an incident Love wave
fundamental mode (mode
This method is founded on representing the field of surface waves as a
superposition of plane homogeneous and inhomogeneous body waves. Body wave
reflection and transmission coefficients are computed over the vertical
interface to obtain outgoing stresses and displacements on the interface.
This outgoing stress-displacement system is regarded as the source of
outgoing surface waves, whose amplitudes are computed via its projection onto
the outgoing surface wave eigenfunctions. In this study the method, hereafter
referred to as the GAl method, is implemented, as follows for Love waves:
SH wave reflection and transmission coefficients over the vertical contact are calculated using the phase velocity of medium
1. Elementary formulae are used for the SH reflection and transmission
coefficients ( These equations are applied individually to each section of the vertical
interface, i.e. each layered contact (there is really a layer index in
addition to the medium index in Eqs. Outgoing stresses and displacements as a function of depth over the vertical interface
are calculated using the eigenfunction of the incident Love wave mode and the SH reflection/transmission coefficients. Coupling between the interface stress-displacement system and the eigenfunctions of
outgoing Love waves (eigenfunctions of medium 1 for reflection, those of medium 2 for transmission)
is calculated by means of the
Since Herrera's scalar product may be computed for any two
stress-displacement vectors
So if
Transmission surface ratios obtained from the SWRT code (solid
lines) and FD calculations (dots), for an incident Love wave fundamental mode
propagating through the modified model L in the forward direction.
Transmission surface ratio obtained by the GF method (solid lines) and FD calculations (dots), for an incident Rayleigh wave fundamental mode propagating through model L in the forward direction. Note that the conversion to higher modes is much weaker than in the Love wave case, but the SWRT result is in good agreement with the FD result.
This method, hereafter referred to as the GF method, has been implemented in
SWRT for both Love and Rayleigh waves. We follow the algorithm of
Love waves — these expressions are easily derived as there is only one component of displacement
and traction relevant for Eq. ( Rayleigh waves — the expressions have been derived by
In our implementation
Schematic representation of the model of
Transmission surface ratios for an incident Love wave fundamental
mode propagating in the forward direction
Finally, we note that in computing the transmission surface ratio (
The three methods described in this section are each implemented as Python
programs in the SWRT code. The only inputs these programs require are the
eigenfunctions for the two media on either side of the lateral discontinuity
and the depths of horizontal interfaces in these media. The programs have
been tested using published results in the literature
(Appendix
We use a single model to demonstrate the capabilities of the GAl and GF
methods. This is a model of an ocean–continent boundary known as “model L”
(Fig.
To validate our results of transmission to multiple higher modes, we compare
the SWRT result with results from finite-difference (FD) simulations. The FD
simulations are performed using the method of
Figure
Figures
This paper has described the author's SWRT code and used it to demonstrate cross-branch mode coupling, of both Love and Rayleigh waves, in simple 2-D models for which previously published results showed only self coupling. SWRT provides a consolidated implementation of sophisticated algorithms by previous authors, which is fully capable of modelling mode conversions arising from normal incidence at a single lateral discontinuity. These algorithms are more accurate than that of Alsop66, whose ability to model mode conversion has been known since the original author's work.
We conclude with remarks on the transferability and flexibility of the SWRT code. Whilst the code has been written to work with a specific format of input file (containing the surface wave eigenfunctions), it is modular with respect to file input: the eigenfunction-reading module of SWRT, which has nothing to do with the algorithms discussed in this paper, can simply be replaced in order to adapt the code to differently formatted input files. Possible additional work may be to modify the expressions for the coupling coefficients in the main programs, depending on the theoretical definitions of the eigenfunctions. As a further comment on flexibility, there is no restriction on the type of normalization that the input eigenfunctions must obey; only the expressions for surface ratios will need modification if the eigenfunctions are not normalized to unit surface displacement. Finally, whilst results have been presented in this paper only for the case of an incident fundamental mode, application of SWRT to the case of arbitrary mode number being incident can be trivially achieved.
SWRT is available at
The Rayleigh wave stress-displacement vector defined by
For the type of problem under consideration, with the plane interface being
normal to the
The first requirement is to obtain an expression for the scalar product
Similarly the expressions for the coupling integrals
Figure
Figure
Rayleigh wave fundamental mode transmission surface ratios for two
types of models taken from ItsYan85:
Figure
Love wave surface transmission ratios
The author declares that he has no conflict of interest.
I would like to acknowledge the Dr. Manmohan Singh Scholarship provided by St. John's College in support of a PhD studentship at the University of Cambridge as well as a Society of Exploration Geophysicists (SEG) Foundation scholarship (SEG/Chevron Scholarship and SEG/John Bookout Scholarship) received during the last year of my study. I would also like to thank Keith Priestley for introducing me to the literature on which this work is based.Edited by: Luis Vazquez Reviewed by: three anonymous referees