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Abstract

We use the Pareto Joint Inversion, together with the Particle Swarm Optimization, to invert the Love and quasi-Rayleigh surface-wave speeds, obtained from dispersion curves, in order to infer the elasticity parameters, mass densities and layer thickness of the model for which these curves are generated. For both waves, we use the dispersion relations derived by Dalton et al. [1]. Numerical results are presented for three angular frequencies, 15 , 60 and 100 s −1 , and for two, five and seven modes, respectively. Comparisons of the model parameters with the values inverted with error-free input indicate an accurate process. If, however, we introduce a 5% error to the input, the results become significantly less accurate, which indicates that the inverse operation, even though stable, is error-sensitive. Correlations between the inverted elasticity parameters indicate that the layer parameters are more sensitive to input errors than the halfspace parameters. In agreement with Dalton et al. [1], the fundamental mode is mainly sensitive to the layer parameters whereas higher modes are sensitive to both the layer and halfspace properties; for the second mode, the results are more accurate for low frequencies.
On Pareto Joint Inversion of guided waves
Adrian Bogacz
, David R. Dalton
, Tomasz Danek
, Katarzyna Miernik§
,
Michael A. Slawinski
December 29, 2017
Abstract
We use the Pareto Joint Inversion together with the Particle Swarm Optimization, to invert the
Love and quasi-Rayleigh surface wave data, which consist of dispersion curves, to obtain the
elasticity parameters, mass densities and layer thickness of the model. For either wave, we use the
dispersion relations derived by Dalton et al. [1]. Results are presented for three angular frequencies,
15,60,100 s1, and for two, five and seven modes, respectively. Comparisons of the input and
inverted values indicate a stable and accurate process.
1 Introduction
Two types of guided waves can propagate in an elastic layer overlying an elastic halfspace (e.g., Dal-
ton et al. [1]). On the surface, their displacements are perpendicular to the direction of propagation.
Though the displacements for both waves are parallel to the surface, the displacement of one of
them is perpendicular to the direction of propagation and, for the other, parallel to that direction.
The former is called the Love wave, and the latter the quasi-Rayleigh wave, where the prefix dis-
tinguishes it from the Rayleigh wave that exists in the halfspace alone and, in contrast to its guided
counterpart, is not dispersive. The orthogonal polarization of the displacement vectors on the sur-
face allows us to identify these waves. Hence, we can use their measurable quantities, such as
propagation speeds, to infer information about the model in which they propagate. In this paper,
Department of Geoinformatics and Applied Computer Science, AGH University of Science and Technology,
Krak´
ow, Poland, abogacz@agh.edu.pl
Department of Earth Sciences, Memorial University of Newfoundland, dalton.nfld@gmail.com
Department of Geoinformatics and Applied Computer Science, AGH University of Science and Technology,
Krak´
ow, Poland, tdanek@agh.edu.pl
§Department of Geoinformatics and Applied Computer Science, AGH University of Science and Technology,
Krak´
ow, Poland, kmiernik@agh.edu.pl
Department of Earth Sciences, Memorial University of Newfoundland, mslawins@mac.com
1
arXiv:1712.09850v1 [physics.geo-ph] 28 Dec 2017
we use the dispersion relations, which involve these speeds and model parameters, and the Pareto
Joint Inversion, to obtain these parameters.
2 Previous work
There are several important contributions to inversion of dispersion relations of guided waves for
model parameters. Let us comment on contributions of particular relevance to our work.
A common technique to obtain quasi-Rayleigh-wave dispersion curves is the Multichannel
Analysis of Surface Waves technique (Park et al., [2]). An approach to inverting such curves for
multiple layers is given in Xia et al. [3], who use the Levenberg-Marquardt and singular-value
decomposition techniques to analyze the Jacobian matrix, and demonstrate sensitivity of material
properties to the dispersion curve.
Wathelet et al. [4] use a neighbourhood algorithm, which is a stochastic direct-search technique,
to invert quasi-Rayleigh-wave dispersion curves obtained from ambient vibration measurements.
Lu et al. [5] invert quasi-Rayleigh waves in the presence of a low-velocity layer, using a genetic
algorithm. Boxberger et al. [6] perform a joint inversion, based on a genetic algorithm, using
quasi-Rayleigh and Love wave dispersion curves and Horizontal-to-Vertical Spectral Ratio curves
obtained from seismic noise array measurements. Fang et al. [7] invert surface wave dispersion data
without generating phase or group velocity maps, using raytracing and a tomographic inversion.
Xie and Liu [8] do Love-wave inversion for a near-surface transversely isotropic structure, using
the Very Fast Simulated Annealing algorithm.
Wang et al. [9] use surface wave phase velocity inversion, based on first-order perturbation
theory, including multiple modes and both quasi-Rayleigh and Love waves, to examine intrinsic
versus extrinsic radial anisotropy in the Earth; the latter anisotropy refers to a homogenized model.
Wang et al. [9] use the classical iterative quasi-Newton method to minimize the L2norm misfit and
introduce the Generalized Minimal Residual Method.
Dal Moro and Ferigo [10] carry out a Pareto Joint Inversion of synthetic quasi-Rayleigh- and
Love-wave dispersion curves for a multiple-layer model using an evolutionary algorithm optimiza-
tion scheme. Dal Moro [11] examines a Pareto Joint Inversion using an evolutionary algorithm of
the combined quasi-Rayleigh and Love wave surface-wave dispersion curves and Horizontal-to-
Vertical Spectral Ratio data. Dal Moro et al. [12] perform a three-target Pareto Joint Inversion
based on full velocity spectra, using an evolutionary algorithm optimization scheme.
3 Dispersion relations
To derive dispersion relations Dalton et al. [1] consider an elastic layer of thickness Zoverlying
an elastic halfspace. Using Cartesian coordinates, we set the surface at x3= 0 , and the interface
at x3=Z, with the x3-axis positive downward. The layer consists of mass density, ρu, and
elasticity parameters, Cu
11 and Cu
44 . The quantities of the halfspace are denoted with superscript d.
2
The same quantities can be expressed in terms of the Pand Swave speeds,
α(·)=sC(·)
11
ρ(·), β(·)=sC(·)
44
ρ(·).
For the Love wave, the dispersion relation is
D`(v`) = det
eιb0
`eιb0
`0
ιsu
`Cu
44 ιsu
`Cu
44 sd
`Cd
44
1 1 1
= 2su
`Cu
44 sin b0
`2sd
`Cd
44 cos b0
`= 0 ,(1)
where
κ`=ω/v`, su
`=sv2
`
(βu)21, sd
`=s1v2
`
(βd)2, b0
`=κ`su
`Z , b`=κ`sd
`Z .
This equation has real solutions, v`, for βu< v`< βd, which are referred to as modes; each
solution can be represented by a dispersion curve of v`plotted against ω, along which D`is zero.
The solution with the lowest value of v`for a given ωis called the fundamental mode.
Formally, the dispersion relation for the quasi-Rayleigh wave is given in terms of the determi-
nant of a 6×6matrix. As shown by Dalton et al. [1] and summarized in Section 4, this can be
reduced to the determinant of a 2×2matrix. For modes other than the fundamental mode, this
equation has a solution only for βu< vr< βd< αdand usually, but not always, for vr> αu
(Ud´
ıas, [13]). For the fundamental mode there is a solution for vr< βu, for higher values of ω.
For vr> αu, the determinant is real; for βu< vr< αu, the determinant is imaginary; for vr< βu
the determinant is real.
4 Pareto Joint Inversion
In this study, the dispersion relations of the Love and quasi-Rayleigh waves are the two target
functions to be examined together by the Pareto Joint Inversion. In general, we search for
min[f1(x), f2(x), . . . , fn(x)] , x S , (2)
where fiare target functions and Sis the space of acceptable solutions. In this study, f1= Drand
f2= D`, given, respectively, in expression (1), above, and expression (4), below. Every solution
is
C={yRn:y=f(x) : xS}.(3)
Among them, a Pareto solution is vector xS, such that all other vectors of this type return a
higher value of at least one of the functions, fi.
The set of all Pareto optimal solutions is Pand PF= (f1(x), f2(x), . . . , fn(x)) , where
x∈ P, is the Pareto front. Each iteration of the algorithm generates a single Pareto solution to be
3
added to the Pareto front. In this paper, the Particle Swarm Optimization algorithm (Kennedy and
Eberhart [14], Parsopoulos and Vrahatis [15]) is used to obtain each element of the Pareto front.
The target function for Love waves is equation (1). Since this is a purely real equation, com-
putations involving imaginary numbers are not necessary.
The 6×6determinant for quasi-Rayleigh waves can be written as (Dalton et al. [1])
Dr(vr)=4Cu
44 det suX suS
ruT ruY,(4)
where
X:= (su)2)1(v2
rq+ 2p)B0+ 2prdcos b0+2 ru(2pv2
rρd) sin a0+rd(2p+v2
rρu) cos a0,
Y:= (su)2)1(v2
rq+ 2p)A02psdcos a0+2 sd(2p+v2
rρu) cos b0su(2pv2
rρd) sin b0,
S:= (su)2)1(v2
rρu+ 2p)sdB0+ (2pv2
rρd) cos b0+2 (2p+v2
rq) cos a0+ 2prusdsin a0,
T:= (su)2)1rd(v2
rρu+ 2p)A0(2pv2
rρd) cos a02(2p+v2
rq) cos b0+ 2surdpsin b0,
with κ, q, p, A0, B0, ru, su, rd, sd, a0, b0, a, b given by κ=ω/vr,q:= ρuρd,p:= Cd
44 Cu
44 ,
A0:=
sin a0
ruru6= 0
κZ ru= 0
, B0:=
sin b0
susu6= 0
κZ su= 0
.
ru=sv2
r
(αu)21, su=sv2
r
(βu)21, rd=s1v2
r
(αd)2, sd=s1v2
r
(βd)2
and
a0=κruZ , b0=κsuZ , a =κrdZ , b =κsdZ .
These equations include several corrections to the formulas on p. 200 of Ud´
ıas [13] (Dalton et
al. [1]). Values of Drcan be imaginary if the product of ruand suis imaginary. We use Dr2=
XY ST as the target function for quasi-Rayleigh waves; it is real for the input values of Cu
11 ,
Cu
44 ,ρu,Cd
11 ,Cd
44 ,ρdand Z. Even in this case, since the parameters do not have any constraints
in the calculations, which involve randomization, complex numbers still appear. However, using
sin(ιx) = ιsinh(x)and cos(ιx) = cosh(x), we restrict our computations to real numbers.
5 Numerical results
Figures 1–3 illustrate Pareto fronts. The three figures are for ω= 15 s1,60 s1and 100 s1,
respectively. The horizontal axes correspond to the misfit of the quasi-Rayleigh dispersion relation
and the vertical axes to the misfit of the Love dispersion relation.
4
(a) (b)
Figure 1: Pareto fronts for ω= 15 s1; plots (a) and (b): first mode and second mode
The values along these axes are to be scaled as follows. Figure 1: ×105; Figure 2 (a, b,
c): ×106; Figure 2 (d, e): ×105; Figure 3 (a): horizontal axis ×109, vertical axis ×105; Fig-
ure 3 (b): horizontal axis ×1011 , vertical axis ×105; Figure 3 (c): horizontal axis ×1010 , ver-
tical axis×105; Figure 3 (d): horizontal axis ×109, vertical axis ×105; Figure 3 (e): ×107;
Figure 3 (f, g): ×106.
In all cases, the fronts have rectangular shape and cover a relativity large range of values,
which means that, while obtaining an optimal value for one target function is relatively easy, the
model parameters producing this value can generate a wide range of values of the other function.
This emphasizes the need for a joint inversion to avoid the ambiguity resulting from inverting
the dispersion relation for a single type of waves. In other words, the redundancy of information
increases the trustworthiness of inferences.
Examining all frequencies and modes, we see that proposed solutions, marked by dark tri-
angles, are more concentrated near (0,0) , along the horizontal axis, which corresponds to quasi-
Rayleigh waves, and more spread out along the vertical axis, which corresponds to the Love waves.
It means that quasi-Rayleigh waves lend themselves particularly well to such an optimization and,
if a satisfactory solution for Love waves is found, the quasi-Rayleigh target function can be easily
adjusted.
Let us examine Table 1, which contains the actual and estimated values, as well as Table 2,
where the ratio of the actual and estimated values is expressed in terms of percents. The values of
all parameters are inverted satisfactorily. As in the case of a few outliers appearing in Figures 1–3,
discrepancies might be caused by differences in positions of global minima of target functions and
by occasional spurious results due to local minima and the numerical complexity the algorithm.
Figures 4–6 are histograms of model parameters obtained along the Pareto fronts for the funda-
mental mode at ω= 60 s1. The spread in values is not due to perturbations, as is commonly the
case for histograms, but instead shows the range of Pareto optimal solutions along the Pareto front.
5
(a) (b)
(c) (d)
(e)
Figure 2: Pareto fronts for ω= 60 s1; plots (a) to (e): first mode to fifth mode
6
(a) (b)
(c) (d)
7
(e) (f)
(g)
Figure 3: Pareto fronts for ω= 100 s1; plots (a) to (g): first mode to seventh mode
8
Table 1: Summary of obtained results: Elasticity parameters are in units of 1010N/m2, mass
densities in 103kg/m3and layer thickness in metres
Cu
11 Cu
44 Cd
11 Cd
44 ρuρdZ
Actual 1.980 0.880 10.985 4.160 2.200 2.600 500.0
ω= 15 s1
2nd mode 2.211 0.875 10.958 4.215 2.234 2.605 522.7
1st mode 1.999 0.893 10.405 3.919 2.254 2.578 480.3
ω= 60 s1
5th mode 2.026 0.919 10.919 4.354 2.240 2.690 512.0
4th mode 2.035 0.893 10.707 4.128 2.256 2.464 494.9
3rd mode 2.022 0.878 11.307 4.258 2.184 2.743 505.1
2nd mode 2.009 0.881 11.239 3.805 2.221 2.667 476.2
1st mode 1.992 0.884 11.030 3.951 2.212 2.632 491.5
ω= 100 s1
7th mode 2.015 0.751 10.089 4.168 2.177 2.724 517.4
6th mode 1.949 0.871 10.848 4.285 2.174 2.563 501.1
5th mode 1.981 0.878 10.846 4.074 2.192 2.682 500.4
4th mode 2.038 0.912 10.683 4.079 2.283 2.561 498.0
3rd mode 1.954 0.872 10.882 4.210 2.174 2.538 506.1
2nd mode 2.046 0.906 10.547 4.334 2.265 2.579 499.9
1st mode 2.123 0.857 11.089 4.315 2.197 2.052 548.8
9
Table 2: Estimated values compared to actual values, in percentages
Cu
11 Cu
44 Cd
11 Cd
44 ρuρdZ
Actual 100.0 100.0 100.0 100.0 100.0 100.0 100.0
ω= 15 s1
2nd mode 111.7 99.4 99.8 101.3 101.6 100.2 104.5
1st mode 100.9 101.5 94.7 94.2 102.5 99.2 96.1
ω= 60 s1
5th mode 102.3 104.4 99.4 104.7 101.8 103.4 102.4
4th mode 102.8 101.5 97.5 99.2 102.6 94.8 99.0
3rd mode 102.1 99.8 102.9 102.4 99.3 105.5 101.0
2nd mode 101.5 100.1 102.3 91.5 101.0 102.6 95.2
1st mode 100.6 100.5 100.4 95.0 100.5 101.2 98.3
ω= 100 s1
7th mode 101.8 85.3 91.8 100.2 99.0 104.8 103.5
6th mode 98.4 99.0 98.8 103.0 98.8 98.6 100.2
5th mode 100.0 99.7 98.7 97.9 99.6 103.1 100.1
4th mode 102.9 103.6 97.3 98.1 103.8 98.5 99.6
3rd mode 98.7 99.1 99.1 101.2 98.8 97.6 101.2
2nd mode 103.3 103.0 96.0 104.2 103.0 99.2 100.0
1st mode 107.2 97.4 100.9 103.7 99.8 78.9 109.8
10
Figure 4: Layer thickness, in metres, and the layer and halfspace mass densities, in 103kg/m3;
black lines represent the actual values
For each parameter there is a good match between the values obtained by the inverse process and
the values used in the original dispersion relations. Also, the aforementioned spread is narrow, as
expected in view of the concentration of solutions near (0,0) , in Figures 1–3. For higher modes
and different frequencies, the results are similar to the ones illustrated in Figures 4–6, as can be
inferred from Tables 1 and 2.
6 Discussion
The obtained results support the use of the Love-wave and quasi-Rayleigh-wave data measured
on the surface in a Pareto Joint Inversion using Particle Swarm Optimization. The inverted model
parameters are accurate and stable without any further constraints.
In a subsequent study, we examine the stability of these parameters under perturbations of input
values. Examining results of the present and subsequent study, from the applied-seismology view-
point, might allow us to adjust data acquisition in such a manner as to enhance the trustworthiness
of inverted parameters.
Solutions presented in this paper are obtained by considering a single mode at a time. A method
that could use several modes simultaneously might be a significant improvement of efficiency.
Acknowledgments
We wish to acknowledge discussions with Piotr Stachura and Theodore Stanoev, as well as the
graphic support of Elena Patarini. This research was performed in the context of The Geome-
11
Figure 5: Layer elasticity parameters, in 1011 N/m2; black lines represent actual values
Figure 6: Halfspace elasticity parameters, in 1011 N/m2; black lines represent actual values
12
chanics Project supported by Husky Energy. Also, this research was partially supported by the
Natural Sciences and Engineering Research Council of Canada, grant 238416-2013, and by the
Polish National Science Center under contract No. DEC-2013/11/B/ST10/0472.
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... Results of this study, in particular empirical adequacy of modelling techniques, allow us to gain insight into the reliability of a joint inverse of the quasi-Rayleigh and Love dispersion curves for obtaining model parameters. A work on that subject is presented by Bogacz et al. [26]. ...
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The dispersion and the surface displacement as a function of frequency of multiple modes guided waves in stratified media including a low-velocity layer are studied by numerical simulation and experiment. A method is developed to determine the thickness and the shear wave velocity of individual layers. First, the modal analysis of Rayleigh wave is investigated numerically for three layered media. Then, ultrasonic surface measurements are performed for three specimens: Steel half-space, Lucite/Steel half-space and Aluminum/Lucite/Steel half-space. The Characteristics of the dispersion curves are analyzed using the frequency-wavenumber method. The non-dispersive Rayleigh wave is obtained for the first simple specimen. The dispersion curves for two modes are obtained for the second specimen with a low-velocity layer on a fast substrate. The dispersion curves for the third specimen containing a low-velocity layer are apparently discontinuous and correspond to different mode branches. Further analysis demonstrates that the apparent discontinuity is caused by a rapid change of mode excitation with frequency at the surface. While one mode vanishes from the recorded wavefield, the other appears. This indicates that the surface displacements of the modes should be also accounted for in the inverse problem, especially in stratified media with a low-velocity layer. Finally, shear wave velocity profiles are inverted based on the experimental (maybe discontinuous) dispersion curves of fundamental or/and higher modes using a Genetic Algorithm(GA). Besides the dispersion characteristics of each mode, the surface displacement distribution is also taken into account for the case of a low-velocity layer, and as a result, the mode-misidentification is avoided.
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This textbook for upper division undergraduates and graduate students provides the ideal introduction to seismology. A student-friendly text fully details the fundamental concepts and includes a step-by-step development of the relevant mathematics. Beginning with clear examples of introductory topics such as one-dimensional problems and liquid media, the book goes on to cover most of the fundamental concepts in seismology. The author describes the application of seismology to the knowledge of the structure of the earth's interior and the origin and nature of earthquakes. Coverage includes seismic wave propagation, normal mode theory, ray theory approximation, body and surface waves, source mechanisms and kinematic and dynamic models. The book also contains appendices on useful mathematical tools and includes extensive problems that help students to understand the basic concepts in this area.
Article
Passive recordings of seismic noise are increasingly used in earthquake engineering to measure in situ the shear-wave velocity profile at a given site. Ambient vibrations, which are assumed to be mainly composed of surface waves, can be used to determine the Rayleigh-wave dispersion curve, with the advantage of not requiring artificial sources. Due to the data uncertainties and the non-lin- earity of the problem itself, the solution of the dispersion-curve inversion is generally non-unique. Stochastic search methods such as the neighbourhood algorithm allow searches for minima of the misfit function by investigating the whole parameter space. Due to the limited number of parame- ters in surface-wave inversion, they constitute an attractive alternative to linearized methods. An efficient tool using the neighbourhood algorithm was developed to invert the one-dimensional Vs profile from passive or active source experiments. As the number of generated models is usually high in stochastic techniques, special attention was paid to the optimization of the forward compu- tations. Also, the possibility of inserting a priori information into the parametrization was intro- duced in the code. This new numerical tool was successfully tested on synthetic data, with and without a priori infor- mation. We also present an application to real-array data measured at a site in Brussels (Belgium), the geology of which consists of about 115 m of sand and clay layers overlying a Palaeozoic base- ment. On this site, active and passive source data proved to be complementary and the method allowed the retrieval of a Vs profile consistent with borehole data available at the same location.