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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 s−1, 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

`−e−ιb0

`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)2−1, sd

`=s1−v2

`

(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={y∈Rn:y=f(x) : x∈S}.(3)

Among them, a Pareto solution is vector x∗∈S, 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 P∗and 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(2p−v2

rρd) sin a0+rd(2p+v2

rρu) cos a0,

Y:= (su)2)−1(v2

rq+ 2p)A0−2psdcos a0+2 −sd(2p+v2

rρu) cos b0−su(2p−v2

rρd) sin b0,

S:= (su)2)−1−(v2

rρu+ 2p)sdB0+ (2p−v2

rρd) cos b0+2 (2p+v2

rq) cos a0+ 2prusdsin a0,

T:= (su)2)−1rd(v2

rρu+ 2p)A0−(2p−v2

rρd) cos a0−2(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:= ρu−ρd,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)2−1, su=sv2

r

(βu)2−1, rd=s1−v2

r

(αd)2, sd=s1−v2

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 s−1,60 s−1and 100 s−1,

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 s−1; plots (a) and (b): ﬁrst mode and second mode

The values along these axes are to be scaled as follows. Figure 1: ×10−5; Figure 2 (a, b,

c): ×10−6; Figure 2 (d, e): ×10−5; Figure 3 (a): horizontal axis ×10−9, vertical axis ×10−5; Fig-

ure 3 (b): horizontal axis ×10−11 , vertical axis ×10−5; Figure 3 (c): horizontal axis ×10−10 , ver-

tical axis×10−5; Figure 3 (d): horizontal axis ×10−9, vertical axis ×10−5; Figure 3 (e): ×10−7;

Figure 3 (f, g): ×10−6.

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 s−1. 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 s−1; 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 s−1; 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 s−1

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 s−1

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 s−1

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 s−1

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 s−1

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 s−1

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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