Geophys. J. Int. (2007) 170, 217–238
doi: 10.1111/j.1365-246X.2007.03383.x
Computation of partial derivatives of Rayleigh-wave phase velocity
using second-order subdeterminants
M. Cercato
D.I.T.S. - Area Geofisica, Università di Roma ‘La Sapienza’, Via Eudossiana 18, 00186 Rome, Italy. E-mail: [email protected]
SUMMARY
Rayleigh-wave propagation in a layered, elastic earth model is frequency-dependent (dispersive) and also function of the S-wave velocity, the P-wave velocity, the density and the thickness of the layers. Inversion of observed surface wave dispersion curves is used in many fields,
from seismology to earthquake and environmental engineering. When normal-mode dispersion curves are clearly identified from recorded seismograms, they can be used as input for a
so-called surface wave ‘modal’ inversion, mainly to assess the 1-D profile of S-wave velocity.
When using ‘local’ inversion schemes for surface wave modal inversion, calculation
of partial derivatives of dispersion curves with respect to layer parameters is an essential and
time-consuming step to update and improve the earth model estimate. Accurate and high-speed
computation of partial derivatives is recommended to achieve practical inversion algorithms.
Analytical methods exist to calculate the partial derivatives of phase–velocity dispersion
curves. In the case of Rayleigh waves, they have been rarely compared in terms of accuracy
and computational speed.
In order to perform such comparison, we hereby derive a new implementation to calculate
analytically the partial derivatives of Rayleigh-mode dispersion curves with respect to the layer
parameters of a 1-D layered elastic half-space. This method is based on the Implicit Function
Theorem and on the Dunkin restatement of the Haskell recursion for the calculation of the
Rayleigh-wave dispersion function. The Implicit Function Theorem permits calculation of the
partial derivatives of modal phase velocities by partial differentiation of the dispersion function.
Using a recursive scheme, the partial derivatives of the dispersion function are derived by a
layer stacking procedure, which involves the determination of the analytical partial derivatives
of layer matrix subdeterminants of order two.
The resulting algorithm is compared with methods based on the more widely used variational theory in terms of accuracy and computational speed.
Key words: elastic-wave theory, Fréchet derivatives, Rayleigh waves, sensitivity.
1 I N T RO D U C T I O N
In surface wave inversion, observed dispersion (variation of phase velocity with frequency) is used to infer the earth’s material properties.
The dispersion characteristics are generally extracted from recorded seismograms in terms of phase–velocity curves of normal modes of
propagation.
The underlying hypothesis to perform this kind of inversion is that of clear mode identification from recorded seismograms.
From classical seismology problems as assessing the Earth’s material properties from surface measurements, as in Dorman & Ewing
(1962), Nolet (1981) and Al-Eqabi & Herrmann (1993), for instance, to high-frequency, near-surface soil characterization as in Gabriels et al.
(1987), Xia et al. (1999) and Park et al. (1999), seismic applications based on surface wave analysis have been developing exponentially and
successfully in the past decades.
Consider a 1-D elastic layered half-space (Fig. 1) as being representative of the subsoil under investigation. This parametric model, made
up by horizontal, elastic, isotropic and homogeneous layers, is bounded above by a free surface and below by a homogeneous isotropic elastic
half-space. The surface wave dispersion equation in such a system can be derived only in implicit form (Aki & Richards 2002), making surface
wave inversion a non-linear inverse problem.
Non-linear problems are often solved using ‘local’ inversion strategies, seeking iteratively for a solution which are ‘linearly close to an
initial guess’ (Menke 1989), making use of Fréchet derivatives (Parker 1994).
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GJI Seismology
Accepted 2007 January 26. Received 2007 January 25; in original form 2006 October 31
218
M. Cercato
x
FREE SURFACE
0
α1, β1, ρ1
z1
h1
zi
αi, βi, ρi
zi+1
hi
zn−1
αn , βn, ρn
z
Figure 1. The plane-layered earth model of reference. Each layer is elastic, homogeneous and isotropic.
In the case of surface wave modal inversion, calculation of partial derivatives of dispersion curves with respect to layer parameters is, at
each iteration, an essential step to update the current earth model estimate.
The number of partial derivatives to be evaluated during the inversion process depends on the number of unknowns (i.e the ‘active’ or
‘running’ inversion parameters), the number of experimental data (i.e. the number of observed dispersion points) and the number of iterations
to be performed. In a broader sense, partial derivatives of modal phase velocities also permits estimation of the ‘sensitivity’ of the dispersion
curves with respect to changes in the layer parameters of the underlying earth model (Julià et al. 2000; Socco & Strobbia 2004).
For a given computer machine, velocity and accuracy of partial derivative calculation for a 1-D earth model depend on several factors:
(1) The algorithm chosen for forward modelling, which includes the calculation of the secular function for the modal phase-velocities and
may also include separate calculation of stresses and displacements at selected depths in the model.
(2) The method selected for partial derivative calculation.
(3) The particular implementation of the numerical algorithms (programming language, numerical routines, etc.).
As far as forward modelling is concerned, modern propagator theory (Gilbert & Backus 1966), introduced by Haskell (1953), has proven to
be a powerful technique for the calculation of the dispersion function. The main drawback of the theory, in its original formulation, is the loss
of accuracy at high frequency, as pointed out by several authors, for example, Schwab & Knopoff (1972), Dunkin (1965), Abo-Zena (1979),
Menke (1979) and Chapman (2003), who provided relevant algorithms to solve the problems.
The Dunkin (1965) restatement of the Haskell–Thompson recursion (Haskell 1953) in terms of second-order subdeterminants is certainly
one of the most popular algorithm for the calculation of Rayleigh wave dispersion function. Successful implementation can be found in Wang
& Herrmann (1980), where the problem is formulated using the notation in Haskell (1964), in Socco & Strobbia (2004), in Watson (1970),
Chouet (1987) and in Wathelet et al. (2004) where this method for forward dispersion-curve calculation is used within the framework of a
global-search inversion procedure.
As the dispersion equation is implicit, forward modelling involves the extraction of its roots by an iterative root-search procedure, which
is the more time-consuming part of the forward surface wave problem.
The methods that have been proposed so far for partial-derivative calculation may be divided into two classes (Novotný 1976).
(1) Numerical differentiation methods (Jeffreys 1961; Takeuchi et al. 1964; Kosloff 1975; Rodi et al. 1975; Xia et al. 1999).
(2) Analytical methods, which may be further divided as follows.
(a) Variational methods (Harkrider 1968; Takeuchi & Saito 1972; Nolet 1981; Lai & Rix 1998; Aki & Richards 2002; Herrmann &
Ammon 2002).
(b) Methods based on the Implicit Function Theorem (Novotný 1970; Schwab & Knopoff 1972; Urban et al. 1993; Novotný et al. 2005).
Numerical differentiation has been developed since the early stages of modern surface wave inversion (Jeffreys 1961; Dorman & Ewing 1962)
and it has been widely applied (Takeuchi et al. 1964; Kosloff 1975; Rodi et al. 1975; Xia et al. 1999) since then.
It is simple in principle. Once that the forward algorithm is established, numerical differentiation allows to evaluate partial derivatives
by repetitive forward modelling calculations, approximating the partial derivatives with finite differences. The shortcoming of the method is
that a great number of direct calculations must be accomplished to keep numerical accuracy under control.
As reported by Novotný (1976), either speed or accuracy of numerical differentiation may be unsatisfactory in calculating surface wave
partial derivatives.
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In this paper, attention is focused on analytical methods to compute dispersion-curve partial derivatives, which can be implemented at
reduced computational cost.
Systematic comparisons between different methods for partial derivative calculation of dispersion curves have been rarely performed. In
the case of Love waves, Novotný (1970) showed that the Implicit Function Theory seems to lead to very accurate results at low computational
costs.
We hereby derive a formulation for calculating the partial derivatives of modal phase velocity based on the Implicit Function Theorem,
adopting the formalism of Dunkin (1965), which leads to rather simple expressions. Partial derivative calculations using variational theory
following Dunkin’s original formulation are also implemented, in order to perform some numerical comparisons between alternative ways for
computing the analytical partial derivatives of Rayleigh-wave phase–velocity dispersion curves.
To improve the readability of the paper, most of the definitions and of the analytical results will be presented in relative Appendices,
leaving in the main body of the paper only those equations which are of practical use to explain significant steps of the theory.
We will first give a brief resume of past developments in analytical partial derivative calculations, then the basics of Dunkin’s theory
will be introduced together with the new formulation of partial derivative calculation. The last part of the paper will be devoted to numerical
testing and comparisons.
2 A N A LY T I C A L M E T H O D S F O R C O M P U T I N G P H A S E V E L O C I T Y PA RT I A L
D E R I VAT I V E S O F S U R FA C E WAV E S
Compared to numerical differentiation, the main merit of analytical methods for computing partial derivatives is that they provide an exact
formulation to the problem, avoiding extensive forward modelling in calculations.
In the notation adopted in this paper, q i is used to indicate a generic physical parameter of layer i, where q i can be either the P-wave
velocity α i , the S-wave velocity β i , the density ρ i or the layer thickness h i .
Partial differentiation of a function g with respect to q i will be indicated as g qi , while the prime mark (i.e. g ) will be used to indicate
partial differentiation with respect to phase velocity c. This should be kept in mind for all the expressions appearing in the following. The
complete set of layer parameters is indicated as Q = (α 1 , ..., α n , β 1 , . . . β n , ρ 1 , . . . , ρ n , h 1 , . . . , h n − 1).
We assume that, for a given earth model, the dispersion curves are the loci where the implicit secular function F R is satisfied:
FR [ω, Q, c(ω, Q)] = 0.
(1)
2.1 The Implicit Function Theorem
From the Implicit Function Theory (Wrede & Spiegel 2002), for a given frequency, the partial derivatives of modal phase velocities with
respect to model parameters can be calculated by partial differentiation of the secular function. Differentiating eq. (1) with respect to the
generic layer parameter q i , we obtain
(FR )qi + FR cqi = 0
(2)
leading to the following identity (Novotný 1976):
(FR )qi
.
(3)
cqi = −
FR
Thus, partial derivatives of the secular function with respect to both the modal phase velocity and the model parameter of interest are
needed to obtain dispersion-curve partial derivatives.
It should be noted that partial differentiation in eq. (3) is performed considering q i and c as being an independent set of variables, without
applying the so-called chain rule.
The implicit function method has been applied to Love wave dispersion in a multilayered model by (Novotný 1970). In the case of
Rayleigh waves, Novotný et al. (2005) calculated partial derivatives of dispersion curves for a simple model of a layer over a half-space,
while Urban et al. (1993) calculated the phase velocity partial derivatives for Rayleigh-wave dispersion function using the forward-problem
implementation by Schwab & Knopoff (1972).
Taking advantage of Dunkin’s formulation, we will develop in Section 3 the analytical expressions for calculating the partial derivatives
of the secular function F R which lead, using eq. (3), to the calculation of dispersion curves partial derivatives.
For a given frequency, two separate recursions must be performed to obtain (FR ) qi and FR , to form the partial derivatives as in
eq. (3).
2.2 VARIATIONAL METHODS
Variational theory is based on Rayleigh principle, which states that kinetic and potential energy of a harmonic wave, averaged over a cycle,
are equal. For a complete review of variational theory in elastodynamics the reader is referred to Kennett (1974).
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Variational theory has been applied by several authors, such as Takeuchi & Saito (1972), Nolet (1981), Wang (1981), Aki & Richards
(2002) and Lai & Rix (1998), for partial-derivative calculations in the case of Rayleigh waves.
Harkrider (1968), Novotný (1970) and more recently Safani et al. (2005) applied variational theory to the calculation of partial derivatives
of Love-wave dispersion curves.
In the Rayleigh-wave case, for the partial derivatives with respect to the S-wave velocity of the generic layer i, it is found that (Lai & Rix
1998):
2
zi ∂c
ρi βi
dr1
dr2
= 2
− 4kr1
kr2 −
dz,
(4)
∂βi
k U I1 zi −1
dz
dz
where r 1 = r 1 (z) and r 2 = r 2 (z) are the real displacement eigenfunctions, U is the group velocity and I 1 , is the first energy integral, defined
as:
∞
I1 =
ρ (r12 + r22 ) dz.
(5)
0
It should be noted that for calculation of group velocity, also the second and third energy integrals are needed (Aki & Richards 2002).
Thus, to obtain the derivatives, it is necessary to calculate the eigenfunctions and their derivatives, the group velocity, the energy integrals
and the integral appearing in the second hand member of eq. (4). An important feature of this method is that, once that the eigenfunctions and
the energy integrals are calculated for a given frequency, partial derivatives with respect to different layer parameters can be obtained with
little numerical effort (basically adding a layer-integral calculation of the type in eq. 4).
3 D U N K I N ’ S T H E O RY A N D N E W F O R M U L AT I O N S
In propagator theory, plane-wave solutions of the equation of motion are propagated through the layer stack, applying the relevant boundary
conditions.
The propagation is handled in terms of displacement–stress amplitude vectors, resulting in square–matrix multiplications of order 4.
Following the formulation of Dunkin (1965), the layer stack matrix R = r i j is defined as:
R = T̄ G n−1 . . . G 1 ,
(6)
where matrices G n−1 , . . . , G 1 are the ‘propagators’ or layer matrices, while T̄ is the inverse of the bottom layer matrix T.
Applying the condition of null stresses at the surface, the secular function can be calculated as a subdeterminant of order 2 of matrix R.
ij
We recall that the second-order subdeterminant p |kl of a square matrix P = p i j of order m > 2, with regard to rows i and j and columns k
and l, is defined as:
ij
p |kl = pik p jl − pil p jk .
(7)
According to Dunkin (1965), the dispersion equation can be stated as:
FR (ω, k) = r11 r22 − r12 r21 = r |12
12 = 0.
(8)
The analytical expressions of the bottom half-space matrix T, of its inverse T̄ and of the generic layer matrix G i appearing in eq. (6), are
reported in Appendix A, where the definitions of the main parameters appearing in the matrix coefficients are introduced.
As the Dunkin’s notation was slightly modified in this paper (for the sake of coherence with current modern notation), a link with the
original parameter definition in Dunkin (1965) is explicitly given in Table A1.
To avoid loss of precision in computation, Dunkin (1965) obtained eq. (8) in terms of second-order subdeterminants of layer matrices,
as:
ef
12
ab
F(ω, k) = r |12
12 = t̄n |ab gn−1 |cd . . . g1 |12 ,
(9)
where T −1 = t̄i j and the summed pairs of indices appearing are to be only distinct pairs of distinct indices.
Using the notation adopted in Watson (1970), F(ω, k) is the first element of the compound matrix of R.
As proved by Dunkin, this formulation if free from the loss of precision related to growing exponential terms. Analytical expression of
matrix subdeterminants involved in the calculation of the secular function are also reported in Appendix A.
Some distinct features of the proposed formulation can be exploited to implement the algorithm efficiently.
12
As pointed out for the first time by Watson (1970), the equality between r i | 12
14 and r i | 23 is preserved throughout the layer stacking, and
12
12
12
12
12
we only need the explicit calculation of r i | 12 , r i | 13 , r i | 14 , r i | 24 and r i | 34 to carry on the recursion layer by layer.
Furthermore, using the identities between subdeterminants appearing in eqs (A33)–(A48) of Appendix A, we don’t need to calculate all
the subdeterminants of the layer matrix to perform the recursion, but only a subset of them.
Finally, as subdeterminants are either pure real or pure imaginary quantities, the layer recursion can be implemented completely using
real algebra.
We can set, for the second-order subdeterminant of a generic layer matrix A = a ij , the following convention:
i j
i j
ij A kl = Im a |kl
(10)
if a kl is imaginary,
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221
14
12
12
so that, for example, G |14
12 = Im(g |12 ) and T̄ |14 = Im(t |14 ). A similar convention is used in Wathelet (2005). Thus, the relevant terms of the
layer stacking recursion for the generic ith step, are:
12
12 12
12 13
12
14
12 24
12 34
ri 12 = ri+1 12 gi 12 + ri+1 13 gi 12 − 2Ri+1 14 G i 12 + ri+1 24 gi 12 + ri+1 34 gi 12
(11)
12
12 12
12 13
12
14
12 24
12 24
ri 13 = ri+1 12 gi 13 + ri+1 13 gi 13 − 2Ri+1 14 G i 13 + ri+1 24 gi 13 + ri+1 34 gi 12
(12)
12
12
12
12
12
13
Ri 14 = Ri 23 = ri+1 12 G i 14 + ri+1 13 G i 14
12 14
12
14
12
14
+ Ri+1 14 2g 14 − 1 + ri+1 24 G i 13 + ri+1 34 G i 12
(13)
12
12 12
12 13
12
13
12 13
12 13
ri 24 = ri+1 12 gi 24 + ri+1 13 gi 24 − 2Ri+1 14 G i 14 + ri+1 24 gi 13 + ri+1 34 gi 12
(14)
12
12 12
12 12
12
12
12 12
12 12
ri 34 = ri+1 12 gi 34 + ri+1 13 gi 24 − 2Ri+1 14 G i 14 + ri+1 24 gi 13 + ri+1 34 gi 12 ,
(15)
where only real terms appear.
The recursion is started at the bottom of the layer stack with the values of subdeterminants of matrix T̄ (see Appendix A).
Using the above eqs (11)–(15), the layer recursion can be efficiently implemented avoiding full matrix multiplication.
In order to calculate the partial derivatives of the secular function appearing in eq. (3), the analytical expressions of derivatives of
layer–matrix subdeterminants, together with the appropriate expressions for the layer recursion, must be derived.
Calculation of partial derivatives of layer matrix subdeterminants, although tedious to derive, is straightforward, and has to be performed
with a certain discipline, developing all terms of partial derivatives which come out from the application of the product differentiation rule.
In Appendix B, we present the partial derivatives of layer-matrix subdeterminants, with respect to c and with respect to the layer parameters
α i , β i , ρ i and h i .
The partial derivatives of the secular function in the right-hand side of eq. (3) can be derived quite efficiently from the partial derivatives
of layer–matrix subdeterminants, once that the recursion in eqs (11) is carefully explored.
When calculating the layer–matrix partial derivative with respect to q i (being α i , β i , ρ i or h i where i indicates the generic layer), it is
easy to observe that only subdeterminants of the layer matrix G i contribute.
In this case, the partial derivative of the dispersion function can be written synthetically as:
12
ab
il
12 e f (FR )qi = r 12 q = t̄n ab gn−1 cd . . . gi gh q . . . g1 12 .
(16)
i
i
On the other hand, when calculating the partial derivative of the secular function with respect to phase velocity, we must consider that the
subdeterminants of all layers depend on c.
Taking eq. (11), using the product differentiation rule, the recursion step in terms of partial derivatives with respect to the modal phase
velocity c, for the generic layer i, is:
12
12 12
gi + ri+1 12 g 12 + r 12 gi 13 + ri+1 12 g 13
ri 12 = ri+1
i+1 13
12
12
12 i 12
12
13 i 12
12
G i 14 − 2Ri+1 12 G 14 + r 12 gi 24 + ri+1 12 g 24
− 2Ri+1
i 12
i+1 24
14
12
14
12
24 i 12
12 34
(17)
gi + ri+1 12 g 34 .
+ ri+1
34
12
34 i 12
Analogous expressions can be obtained for the remaining subdeterminants of R appearing in eqs (12)–(15).
Adding recursively every layer contribution, as per the evaluation of the secular function, the partial derivatives of the secular function
which appear in the second hand member of eq. (3) can be easily calculated at every frequency, for every mode, substituting the modal phase
velocity value of the mode of interest in the analytical expressions of the terms appearing in eqs (16) and (17).
A similar procedure can be adopted for differentiation with respect to ω, which can be used for group velocity calculations (Schwab
& Knopoff 1972). For simplicity, we do not report the analytical expressions of layer matrix subdeterminants with respect to ω: they can be
calculated by the reader in an analogous way as for the other parameters, using simple differentiation with respect to ω.
An algorithm for computing dispersion-curve partial derivatives using variational theory was also implemented, characterized by the
following features.
(i) Normalized stress–displacement eigenfunctions are calculated at layer interfaces using products of propagator and compound matrices,
as described in Harkrider (1979) and Harkrider (1970) and implemented in Herrmann & Ammon (2002).
(ii) Derivatives of stress–displacement eigenfunctions dr 1 /dz and dr 2 /dz are calculated analytically exploiting the following relations
(Aki & Richards 2002).
dr1
r3
= k r2 +
(18)
dz
μ
dr2
r4
kλ
=−
r1 +
.
(19)
dz
λ + 2μ
λ + 2μ
(iii) Integration is performed analytically, as described in Wang (1981), using up and downgoing potential amplitudes at layer interfaces.
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Table 1. Structural parameters of two reference models chosen for algorithm comparison.
Layer
number
α
(km s−1 )
β
(km s−1 )
ρ
(t m−3 )
1
2
3
4
5
6
Model MODX six-layer model in Xia et al. (1999)
0.650
0.194
1.82
0.750
0.270
1.86
1.400
0.367
1.91
1.800
0.485
1.96
2.150
0.603
2.02
2.800
0.740
2.09
1
2
Model MODN 2-layer model in Novotný et al. (2005)
6.00
3.50
2.70
8.00
4.50
3.30
h
(km)
2.0 × 10−3
2.3 × 10−3
2.5 × 10−3
2.8 × 10−3
3.2 × 10−3
∞
35
∞
We end this section using two examples of complete partial derivatives calculation for two earth models, reported in Table 1, taken from
previously published papers. The two methods described in this section give identical results.
In Fig. 2, we report the values of partial derivatives of model MODN in Novotný et al. (2005) with respect to layer parameters.
The values of partial derivatives of model MODX in Xia et al. (1999) are given in Fig. 3 with respect to the S-wave velocities, in Fig. 4
with respect to the P-wave velocities, in Fig. 5 for the densities and finally in Fig. 6 for the layer thicknesses.
As Fig. 2 in Xia et al. (1999) shows, fundamental-mode dispersion curve is almost insensitive to changes in P-wave velocity, especially
in the 15–20 Hz frequency band.
Please note that, to compare the values of partial derivatives of different layer parameters (Novotný et al. 2005) in terms of sensitivity, the
values of partial derivatives should be normalized by the value q i /c, in order to obtain a dimensional quantities (Aki & Richards 2002). If this
calculation is made, it can be observed that sensitivities with respect to S-wave velocity are usually larger than the other partial derivatives.
Fig. 5 shows the relative partial derivatives for model MODX with respect to S-wave velocities and thicknesses of model layers.
The main features of the proposed approaches are as follows.
(i) Calculation of partial derivatives is analytical in its formulation and implementation for both methods, it is thus exact (in the sense that
the numerical accuracy depends only on machine precision).
(ii) Calculation of eigenfunctions can be completely avoided using the Implicit Function Theorem.
(iii) Integration using variational theory is performed analytically using complex algebra.
(iv) Once that group velocity and energy integrals are calculated, variational methods involves only little calculations (a layer integral)
for evaluating derivatives with respect to different layer parameters, while, if the implicit function theorem is used, an additional recursion
through the entire layer stack must be accomplished for each layer parameter.
0.8
∂ c/ ∂ β2
∂ c/ ∂ β1
0.7
0.6
0.5
0.4
0.3
0.2
∂ c/ ∂ α2
0.1
∂ c/ ∂ α1
∂ c/ ∂ ρ 2
0
∂ c/ ∂ h1
−0.1
−0.2
0
∂ c/ ∂ ρ 1
0.02
0.04
0.06
0.08
Frequency (Hz)
0.1
0.12
Figure 2. Model MODNR (Novotný et al. 2005)—partial derivatives of the fundamental-mode phase velocity with respect to layer parameters.
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1.4
1.2
∂ c/ ∂ β6
∂ c/ ∂ β1
1
0.8
∂ c/ ∂ β2
∂ c/ ∂ β4
0.6
0.4
∂ c/ ∂ β3
0.2
0
∂ c/ ∂ β5
−0.2
0
5
10
15
20
25
Frequency (Hz)
30
35
40
Figure 3. Model MODX (Xia et al. 1999)—partial derivatives of the fundamental-mode phase velocity with respect to layer S-wave velocities.
0.05
∂ c/ ∂ α2
0.04
0.03
0.02
∂ c/ ∂ α1
∂ c/ ∂ α6
∂ c/ ∂ α3
0.01
0
∂ c/ ∂ α5
−0.01
∂ c/ ∂ α4
−0.02
0
5
10
15
20
25
Frequency (Hz)
30
35
40
Figure 4. Model MODX (Xia et al. 1999)—partial derivatives of the fundamental-mode phase velocity with respect to layer P-wave velocities.
4 NUMERICAL TESTS
In this section, we aim to compare alternative algorithms for partial derivative calculations in terms of accuracy and computational speed.
We have tested accuracy of computations comparing previously published numerical results (when available), our algorithms (described
c
in the previous section) and two public-domain codes: the Fortran code included in Herrmann & Ammon (2002) and the Matlab
code by
Lai & Rix (1998).
In the following, compared algorithms will be indicated with an acronym, composed of three letters representing sequentially: the initial
of the first author’s surname, the method adopted for partial derivative calculation (V for variational and I for Implicit Function Theorem) and
c
the programming language (C for ANSI C, F for Fortran and M for Matlab
).
Overall, the following algorithms will be compared.
(1) CIC (Cercato, this paper) is the approach proposed in this paper using Implicit Function Theory, Dunkin restatement and Brent method
c
(Press et al. 1997) for root search, originally programmed in ANSI C and then translated into Matlab
(CIM) and Fortran (CIF) for comparison
with third party software.
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M. Cercato
0.06
∂ c/ ∂ ρ 3
∂ c/ ∂ ρ 4
∂ c/ ∂ ρ 6
∂ c/ ∂ ρ 2
0.04
0.02
0
−0.02
∂ c/ ∂ ρ 5
−0.04
−0.06
−0.08
−0.1
∂ c/ ∂ ρ 1
−0.12
−0.14
0
5
10
15
20
25
Frequency (Hz)
30
35
40
Figure 5. Model MODX (Xia et al. 1999)—partial derivatives of the fundamental-mode phase velocity with respect to layer densities.
0
∂ c/ ∂ h5
∂ c/ ∂ h4
∂ c/ ∂ h2
−50
∂ c/ ∂ h3
−100
∂ c/ ∂ h1
−150
0
5
10
15
20
25
Frequency (Hz)
30
35
40
Figure 6. Model MODX (Xia et al. 1999)—partial derivatives of the fundamental-mode phase velocity with respect to layer thicknesses.
(2) CVC (Cercato, this paper). This algorithm takes advantage of variational theory for partial derivative calculation, uses Dunkin restatement, Brent method for root search, and uses analytical integration as in (Wang 1981). It was originally programmed in ANSI C and then
c
translated into Matlab
(CVM) for comparison with third party software.
(3) HVF (Herrmann & Ammon 2002) calculates the partial derivatives using variational technique (programmed in FORTRAN), uses
Dunkin restatement following the notation by Haskell (1964), adopts the Neville method for root-searching and analytical integration Wang
(1981).
(4) LVM (Lai & Rix 1998) uses the variational method to calculate the partial derivatives, the algorithm by Hisada (1994) for secular
c
function calculation, and built-in Matlab
functions for root-finding and numerical integration.
Since all the algorithms work in double precision, comparison in terms of accuracy can be performed in a straightforward way. Accuracy has
been tested extensively calculating, with all the algorithms cited above, the partial derivatives with respect to the S-wave layer velocities for
more than 50 earth-models.
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Journal compilation Computation of partial derivatives of Rayleigh-wave phase velocity
β4 ∂ c
c ∂ β4
1
0.8
0.6
β3 ∂ c
c ∂ β3
225
β1 ∂ c
c ∂ β1
β2 ∂ c
c ∂ β2
β6 ∂ c
c ∂ β6
β5 ∂ c
c ∂ β5
0.4
0.2
0
−0.2
−0.4
−0.6
h2 ∂ c
c ∂ h2
h5 ∂ c
c ∂ h5
h4 ∂ c
c ∂ h4
−0.8
h3 ∂ c
c ∂ h3
−1
0
5
10
15
20
25
Frequency (Hz)
h1 ∂ c
c ∂ h1
30
35
40
Figure 7. Model MODX (Xia et al. 1999)—relative or normalized partial derivatives of the fundamental-mode phase velocity with respect to layer S-wave
velocities (black) and thicknesses (gray).
Table 2. Numerical calculation for the two-layer model MODN in Novotný
et al. (2005).
Frequencies (Hz):0, 025, 0, 0333̄, 0.05
Published⎛results Novotný et al.
⎞ (2005)
0.14381 0.59832
⎝0.33374 0.47141⎠
0.74250 0.15013
HVF (Herrmann
& Ammon⎞2002)
⎛
0.14381 0.59832
⎝0.33374 0.47141⎠
0.74250 0.15013
LVM
⎛ (Lai & Rix 1998)
⎞
0.13934 0.59687
⎝0.32268 0.47113⎠
0.73557 0.14983
CIC, CIM, CVC,
⎛ CVM, CIF (Cercato,
⎞ this paper)
0.14381 0.59832
⎝0.33374 0.47141⎠
0.74250 0.15013
As an example, for the MODN and MODX earth models described in Table 1 we report, in Tables 2 and 3, respectively, the numerical
results of partial-derivative calculations with respect to layer S-wave velocities, in terms of the Jacobian:
∂c Ji j =
.
(20)
∂β j f = fi
The numerical values of the Jacobian appearing in the original papers are also displayed. It can be seen that only the method by Lai & Rix
(1998) shows a certain lack of accuracy, which depends mainly on the fact that numerical integration (1000 points per model) is used to
calculate the energy and layer integrals.
We can conclude that all the proposed methods are capable to give accurate partial derivatives to the aim of sensitivity calculation and
surface wave inversion.
Numerical comparisons must be made more carefully when coming to computational speed.
Different testing condition must be accomplished to allow for a ‘fair’ comparison between different algorithms. We preferred to leave
third party software ‘as it is’ while modify our algorithms for the particular case under investigation.
Particular attention was given to compare algorithms computing the same quantities, using similar numerical implementation (basically
in terms of programming code and I/O formats) as will be described later on.
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M. Cercato
Table 3. Numerical calculation for the six-layer model MODX in (Xia et al.
1999)
Frequency (Hz):[5, 10, 15, 20, 25, 30]
Published results in (Xia et al. 1999)
⎛
⎞
0.018 0.018 0.022 0.021 0.017 0.872
⎜0.130 0.106 0.062 0.025 0.022 0.766⎟
⎜
⎟
⎜1.067 0.925 0.313 0.034 0.017 0.262⎟
⎜
⎟
⎜0.155 1.037 0.967 0.457 0.145 0.040⎟
⎜
⎟
⎝0.293 1.072 0.517 0.102 0.012 0.001⎠
0.520
⎛
0.01809
⎜0.13002
⎜
⎜1.06737
⎜
⎜0.15460
⎜
⎝0.29284
0.52038
0.923 0.202 0.016 0.000 0.000
HVF (Herrmann & Ammon 2002)
⎞
0.01834 0.02219 0.02036 0.01750 0.87242
0.10646 0.06175 0.02467 0.02225 0.76580⎟
⎟
0.92456 0.31290 0.03356 0.01667 0.26222⎟
⎟
1.03665 0.96729 0.45739 0.14507 0.04024⎟
⎟
1.07203 0.51690 0.10261 0.01137 0.00074⎠
0.20184 0.01590 0.00060 0.00001
LVM (Lai & Rix 1998)
⎛
⎞
0.01443 0.01300 0.01383 0.01193 0.01382 0.86939
⎜0.12161 0.08636 0.04675 0.02314 0.01812 0.76576⎟
⎜
⎟
⎜0.95916 0.84527 0.28894 0.02847 0.01531 0.25958⎟
⎜
⎟
⎜0.14780 0.96370 0.92731 0.45258 0.13932 0.03864⎟
⎜
⎟
⎝0.28708 1.04603 0.49245 0.10321 0.01138 0.00070⎠
0.51321 0.89495 0.20270 0.01552 0.00060 0.00001
CIC, CVC, CIM, CVM, CIF (Cercato, this paper)
⎛
⎞
0.01809 0.01834 0.02219 0.02036 0.01750 0.87242
⎜0.13002 0.10646 0.06174 0.02467 0.02225 0.76580⎟
⎜
⎟
⎜1.06766 0.92490 0.31304 0.03359 0.01665 0.26204⎟
⎜
⎟
⎜0.15460 1.03665 0.96729 0.45739 0.14507 0.04024⎟
⎜
⎟
⎝0.29284 1.07203 0.51690 0.10261 0.01137 0.00074⎠
0.52024
0.92342
0.92354
0.20196
0.01592
0.00060
0.00001
Table 4. The two-layer model (engineering scale) chosen for algorithm comparison.
Layer
number
1
2
α
(km s−1 )
β
(km s−1 )
ρ
(t m−3 )
h
(km)
0.60
2.00
0.20
0.80
2.00
2.20
0.030
∞
We report hereby the results related to a single earth model composed by two layers of different physical properties, as per Table 4. This
model has been ‘overparametrized’ up to twenty layers, adding fictitious layers to monitor the execution time taken by the above described
algorithms. For all the following tests, we report the results averaged over at least four repetitions.
Calculation of modal phase velocities is excluded from numerical comparison: algorithms for partial derivative calculations will be
compared for a given set of previously calculated phase velocities.
First, our algorithms CIC, CVC, CIM and CVM are compared, testing a single run calculating the S-wave velocity Jacobian for given
fundamental-mode phase velocities at 100 frequency points.
We display in Fig. 8 the curves of the normalized execution time (of the variational method with respect to the time taken by the implicit
function method) versus the layer number. The behaviour showed by the curves is very different. This is because no intrinsic double complex
type are present in ANSI C, and thus the use of complex algebra has a significant impact on computational speed as complex operation must
be implemented with made-to-purpose functions, causing an increase with layer number of the ratio between execution times of CVC and
CIC. This is a behaviour depending on the programming language.
In fact, the curve representing the ratio between the execution times of CVM and CIM codes shows that the efficiency of the implicit
function method versus the variational method decreases with the number of layers. This is because a complete recursion through the entire
layer stack must be accomplished, for each layer parameter, to calculate the numerator of eq. (3).
In a second test, we compared our codes to the LVM code (Lai & Rix 1998). As LVM code performs partial derivatives calculation for
both P- and S-wave velocities in a single routine, we had to modify our codes accordingly, to make the running time comparable.
The results of the comparison are reported in Fig. 9 (where the ordinate axis is logarithmic). The LVM code is slower because of numerical
integration. To get accurate results, this normally implies calculation of eigenfunctions at thousands of depth points within the earth model.
On the contrary, using analytical integration, eigenfunctions are calculated only at layer interfaces.
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227
7
6
CV C
CI C
Normalized execution time
5
4
3
CV M
CI M
2
1
0
2
4
6
8
10
12
14
16
18
20
Number of Layers
Figure 8. Numerical comparison between the ANSI C and Matlab codes described in this paper. The CIC and CIM codes use the Implicit Function Theorem
while the CVC and CVM codes use variational theory for jacobian calculation.
1000
Normalized execution time
100
LV M
CI M
10
CV M
CI M
1
0.1
2
4
6
8
10
12
14
16
18
20
Number of Layers
Figure 9. Numerical comparison between the Matlab codes CIM and CVM developed in this study and the Matlab code LVM by Lai & Rix (1998).
When comparing the CVM and CIM code, the behaviour is similar to Fig. 8, apart that the CIM code efficiency is reduced if compared
to the CVM code, due to the presence of an additional recursion for P-wave velocity partials.
In a further step we compared a Fortran implementation of the implicit algorithm (CVF) against the Fortran code by Herrmann & Ammon
(2002) that we indicated as HVF.
We adopted the same I/O structures and tested the software for a given subset of one hundred frequency points and modal phase velocities
(again the fundamental mode), calculating in a single routine both the S wave and layer-thickness partial derivatives.
The results of this testing are displayed in Fig. 10. The behaviour is similar to the CVM/CIM curve in Fig. 9. Little differences are due
to the fact that the HVF code adopts the notation in Haskell (1964), which leads to simpler expressions of the propagator matrices (all real
terms) and of the bottom layer matrix (which is real if the phase velocity is less than the S-wave velocity of the half-space).
5 C O N C LU S I O N S A N D F U T U R E WO R K
A new formulation for the calculation of partial derivatives with respect of any structural parameter of a 1-D layered elastic model has been
performed, using the Implicit Function Theorem and second-order subdeterminants development as described in Dunkin (1965).
C 2007 The Author, GJI, 170, 217–238
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3
Normalized execution time
2.5
2
HVF
CIF
1.5
1
0.5
0
2
4
6
8
10
12
14
16
18
20
Number of Layers
Figure 10. Numerical comparison between the Fortran version (CVF) of the Implicit Function code and the the Fortran code HVF by Herrmann & Ammon
(2002).
Using the same notation, the more widely used variational theory was utilized to build another analytical algorithm for numerical
comparisons.
The implemented algorithms, have proven to be stable and accurate, giving equivalent results in terms of accuracy. The main difference
between the implicit and variational approach is that the latter implies calculation of eigenfunctions and energy integrals, which can be further
used for seismogram synthesis.
When coming to speed of computation, the efficiency of the method taking advantage of the Implicit function theorem decreases with
layer number, as partials derivatives are obtained using recursions involving all the layer stack.
Within the framework of modal surface wave inversion for near surface characterization, where earth models made up by few layers are
usually involved, the implicit function method can be used with maximum efficiency.
As the calculation of the jacobian (i.e. the matrix of partial derivatives with respect to model parameters) is one of the more time
consuming part of a single step in a ‘local’ inversion procedure for surface wave dispersion curves, the proposed algorithms may constitute a
valuable tool to speed up conventional surface wave linearized inversion algorithms.
AC K N OW L E D G M E N T S
I am indebted to Prof Ettore Cardarelli and Prof Marcello Bernabini for their careful review of this manuscript before submission, their
criticism and helpful suggestions.
The comments of an anonymous reviewer are also acknowledged.
Prof David Harkrider kindly provided the original slides of the work cited in Harkrider (1979), together with very useful advises to
calculate eigenfunctions and perform integration analytically using potential amplitudes.
Prof Robert Herrmann suggested major improvements to the variational method for calculating partial derivatives, pointing out the way
to obtain the energy and layer integrals analytically. I greatly appreciate his thoughtful and enlightening suggestions that improved this paper
very much.
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A P P E N D I X A : A R E V I E W O F D U N K I N ’ S T H E O RY
If we recall that, in each homogeneous elastic layer, the vertical wavenumbers are defined as:
1
ω2 2
να = k 2 − 2
α
1
ω2 2
νβ = k 2 − 2
β
with α and β being, respectively, the P- and the S-wave layer velocities. Introducing:
⎧
⎨
⎪ ω2 − ω2
if
c<α
2
ω
c2
α2
2
m α = k − 2 = 2
2
⎪
α
⎩ ω2 − ω2
if
c>α
α
c
⎧
ω2
⎪
2
ω ⎨ c2 −
m β = k 2 − 2 = ⎪
β
⎩ ω22 −
β
ω2
β2
if
c<β
ω2
c2
if
c>β
we can derive expression of the vertical wavenumbers which are real valued or pure imaginary.
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(A2)
(A3)
(A4)
M. Cercato
230
In the bottom half-space, due to the radiation condition:
να = m α
(A5)
νβ = m β .
(A6)
In the other layers:
if
mα
να =
i mα
if
νβ =
mβ
i mβ
if
if
c<α
c>α
(A7)
c<β
c>β
(A8)
Following Dunkin (1965), we define:
⎧ k
⎨ m α sinh(m α h)
k
sinh(να h) =
Sα =
⎩ mkα sin(m α h)
να
⎧ k
⎨ m β sinh(m β h)
k
sinh(νβ h) =
Sβ =
⎩ mk sin(m β h)
νβ
β
Cα = cosh(να h) =
Cβ = cosh(νβ h) =
if
c<α
if
c>α
if
c<β
if
c>β
⎧
⎨ cosh(m α h)
if
c<α
⎩ cos(m α h)
if
c>α
⎧
⎨ cosh(m β h)
if
c<β
⎩ cos(m β h)
if
c>β
(A9)
(A10)
(A11)
(A12)
2β 2
c2
να
ν̄α =
k
νβ
ν̄β =
k
γ =
(A13)
(A14)
(A15)
ω2
.
(A16)
β2
Relationships between the parameters appearing in our notation and the ones in the original paper by Dunkin are summarized in
Table A1.
Expressions for the components of the layer matrix G i are:
l = 2k 2 −
g11 = g44 = γ Cα + (1 − γ )Cβ
g12 = g34 = i (1 − γ )Sα + γ ν̄β2 Sβ
g13 = g24 = −i
g14 =
(A17)
(A18)
1
(Cα − Cβ )
ρωc
(A19)
1
(Sα − ν̄β2 Sβ )
ρωc
(A20)
g21 = g43 = i −γ ν̄α2 Sα − (1 − γ )Sβ
(A21)
Table A1. Table linking our notation with Dunkin’s. The D subscript is
intended to indicate the parameter as defined in the original paper.
ξD =k
h D = να
CH D = C α
λ D = iω k −1
h̄ D = ν̄α
SH D = S α
s D = iω
k D = νβ
CK D = C β
l D = 2k 2 −
ω2
β2
γ D = −γ
k̄ D = ν̄β
SK D = S β
= k 2 + νβ2 = l
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2007 The Author, GJI, 170, 217–238
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Journal compilation Computation of partial derivatives of Rayleigh-wave phase velocity
g22 = g33 = (1 − γ )Cα + γ Cβ
g23 = −
(A22)
1 2
ν̄α Sα − Sβ
ρωc
(A23)
g31 = g42 = iρωc γ (1 − γ )(Cα − Cβ )
g32 = −ρωc (1 − γ )2 Sα − γ 2 ν̄β2 Sβ
g41 = −ρωc −γ 2 ν̄α2 Sα + (1 − γ )2 Sβ ,
(A24)
(A25)
(A26)
where the layer-index is dropped for simplicity from the layer parameters.
the bottom layer matrix T and its inverse T −1 = T̄ are, respectively:
⎛
⎞
ik
νβ
ik
−νβ
⎜
⎟
⎜ να
⎟
ik
−να
ik
⎟
T =⎜
⎜
⎟
μl
−2 i kμνβ ⎠
⎝ μ l 2 i kμνβ
−μ l
2 i kμνα
⎛
T̄ = −
β2
2μνα νβ ω2
−2 i kμνα
2ikμνα νβ
⎜
⎜ −μlνα
⎜
⎜ 2iμkν ν
⎝
α β
μlνα
(A27)
−μ l
μlνβ
να νβ
2iμkνα νβ
ikνα
−μlνβ
να νβ
2iμkνα νβ
−ikνα
ikνβ
⎞
⎟
−να νβ ⎟
⎟.
−ikνβ ⎟
⎠
(A28)
−να νβ
The second-order subdeterminants of T̄ which are of use in forward calculations, are:
2
12
β4
4ω2
l
−
t̄ 12 =
4 ω4 να νβ
c2
12
t̄ 13 = −
231
1
4ρω2 νβ
(A29)
(A30)
β2
l
−
2
4 ρ ω3 c να νβ
1
ω2
=
−1 .
4ρ 2 ω4 c2 να νβ
12
12
t̄ 14 = t̄ 23 = i
(A31)
12
t̄ 24
(A32)
Starting from (A17) to (A26), the following second-order layer–matrix subdeterminants are derived:
12
34
g = g = 2γ (1 − γ ) + (2γ 2 − 2γ + 1)Cα Cβ − (1 − γ )2 + γ 2 ν̄ 2 ν̄ 2 Sα Sβ
(A33)
24
12
g 13 = g 34 =
(A34)
12
α β
34
1 Cα Sβ − ν̄α2 Sα Cβ
ρωc
12
14
23
12
g 14 = g 23 = g 34 = g 34 = i
1 (1 − 2γ )(1 − Cα Cβ ) + 1 − γ − γ ν̄α2 ν̄β2 Sα Sβ
ρωc
14
23
12
13
12
g 23 = g 34 = g 34 = g 24 = g 34 =
1 2
ν̄β Cα Sβ − Sα Cβ
ρωc
12
g 34 = −
1
2(1 − Cα Cβ ) + 1 + ν̄α2 ν̄β2 Sα Sβ
ρ 2 ω2 c2
34
13
g 12 = g 24 = ρωc γ 2 ν̄β2 Cα Sβ − (1 − γ )2 Sα Cβ
(A35)
(A36)
(A37)
(A38)
24
13
g 13 = g 24 = Cα Cβ
(A39)
13
13
14
23
g 14 = g 23 = g 24 = g 24 = i (1 − γ )Sα Cβ + γ ν̄β2 Cα Sβ
(A40)
13
g 24 = −ν̄β2 Sα Sβ
(A41)
23
34
34
14
g 12 = g 12 = g 14 = g 23
= iρωc (3γ 2 − 2γ 3 − γ )(1 − Cα Cβ ) + (1 − γ )3 − γ 3 ν̄α2 ν̄β2 Sα Sβ
(A42)
23
24
24
14
g 13 = g 13 = g 14 = g 23 = −i (1 − γ ) Cα Sβ + γ ν̄α2 Sα Cβ
(A43)
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Journal compilation M. Cercato
232
14
23
g 14 = g 23 = 1 − 2 γ (1 − γ ) (1 − Cα Cβ ) + (1 − γ )2 + γ 2 ν̄α2 ν̄β2 Sα Sβ
(A44)
23
14
14
g 23 = g 14 = g 14 − 1
(A45)
34
24
g 12 = g 13 = ρωc (1 − γ )2 Cα Sβ − γ 2 ν̄α2 Sα Cβ
(A46)
24
g 13 = −ν̄α2 Sα Sβ
(A47)
34
g 12 = −ρ 2 ω2 c2 2γ 2 (1 − γ )2 (1 − Cα Cβ ) + [(1 − γ )4 + γ 4 ν̄α2 ν̄β2 ]Sα Sβ .
(A48)
A P P E N D I X B : A N A LY T I C A L F O R M U L A S O F PA RT I A L D E R I VAT I V E S
First, we report the non-zero partial derivatives of the auxiliary functions introduced in Appendix A:
⎧ ω2
if
c<α
⎨ − m α c3
m α =
2
ω
⎩ m c3
if
c>α
α
m αα =
⎧ ω2
⎨ m α α3
⎩−m
ω2
αα
3
⎧
ω2
⎪
⎨ − m β c3
m β =
ω
⎪
⎩ m β c3
2
m ββ =
⎧ 2
ω
⎪
⎨ mβ β3
ω
⎪
⎩ − mβ β3
2
if
c<α
if
c>α
if
c<β
if
c>β
if
c<β
if
c>β
(B1)
(B2)
(B3)
(B4)
2 2c
ν̄α = − 2
α
(B5)
2
2c2
ν̄α α = 3
α
(B6)
2 2c
ν̄β = − 2
β
(B7)
2
2c2
ν̄β β = 3
β
(B8)
γ = −
γβ =
(B9)
4β
c2
l = −
lβ =
4β 2
c3
(B10)
4ω2
c3
(B11)
2ω2
β3
Cα = −
Cαα =
Cαh =
(B12)
kh
Sα
c
(B13)
ωch
Sα
α3
⎧
⎨ m α sinh(m α h)
if
c<β
⎩ −m α sin(m α h)
if
c>β
(B14)
(B15)
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2007 The Author, GJI, 170, 217–238
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Journal compilation Computation of partial derivatives of Rayleigh-wave phase velocity
Cβ = −
Cββ =
Cβh =
kh
Sβ
c
if
c<β
−m β sin(m β h)
if
c>β
Sαα =
(B16)
ωch
Sβ
β3
m β sinh(m β h)
Sα = −
m α
1
+
mα
c
Sα +
(B17)
k h m α
Cα
mα
m αα
(k h Cα − Sα )
mα
Sαh = k Cα
mβ
k h m β
1
Sβ = −
Sβ +
+
Cβ
mβ
c
mβ
Sββ =
233
m ββ
(k h Cβ − Sβ )
mβ
Sβh = k Cβ .
(B18)
(B19)
(B20)
(B21)
(B22)
(B23)
(B24)
We next report the partial derivatives of the layer–matrix subdeterminants with respect to phase velocity and structural parameters.
B1 Phase velocity c
If we set:
δCC = (Cα Cβ + Cα Cβ )
(B25)
δ SS = (Sα Sβ + Sα Sβ )
(B26)
δC S = (Cα Sβ + Cα Sβ )
(B27)
δ SC = (Sα Cβ + Sα Cβ )
(B28)
δν̄ = ν̄α2 ν̄β2 + ν̄α2 ν̄β2 .
(B29)
The expressions of the partial derivatives with respect to phase velocity of the layer matrix subdeterminants which are relevant for dispersion
function computation are:
12
gi 12 = 2γ (1 − 2γ )(1 − Cα Cβ ) + (2γ 2 − 2γ + 1) δCC +
(B30)
− 2γ γ − 1 + γ ν̄α2 ν̄β2 + γ 2 δν̄ Sα Sβ − (1 − γ )2 + γ 2 ν̄α2 ν̄β2 δ SS
12
12
gi 1 δC S − (ν̄α2 ) Sα Cβ − ν̄α2 δ SC
gi 13 = − 13 +
c
ρωc
12
G i 14
1 12
14
12
+
(2γ − 1) δCC − 2γ (1 − Cα Cβ ) +
G i 14 = G i 23 = G i 34 = −
c
ρωc
1 2 2
+
1 − γ − γ ν̄α ν̄β δ SS − γ + γ ν̄α2 ν̄β2 + γ δν̄ Sα Sβ
ρωc
12
gi 24
1 2 12
13
+
ν̄β Cα Sβ + ν̄β2 δC S − δ SC
gi 24 = gi 34 = −
c
ρωc
12
12
2 gi 34
1
+ 2 2 2 2δCC − δν̄ Sα Sβ − 1 + ν̄α2 ν̄β2 δ SS
gi 34 = −
c
ρ ω c
13
13
gi 12
+ ρωc 2γ γ ν̄β2 Cα Sβ + γ 2 ν̄β2 Cα Sβ + γ 2 ν̄β2 δC S +
gi 12 =
c
+ρωc 2γ (1 − γ )Sα Cβ − (1 − γ )2 δ SC
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Journal compilation (B31)
(B32)
(B33)
(B34)
(B35)
M. Cercato
234
13
gi 13 = δCC
(B36)
13
G i 14 = −γ Sα Cβ + (1 − γ ) δ SC + γ ν̄β2 + γ ν̄β2 Cα Sβ + γ ν̄β2 δC S
(B37)
13
gi 24 = − ν̄β2 Sα Sβ + ν̄β2 δ SS
(B38)
14
14
23
G i 12
G i 12 = G i 12 =
+ ρωc γ (−6γ 2 + 6γ − 1)(1 − Cα Cβ ) − (γ 2 − γ )(1 − 2γ ) δCC +
c
+ ρωc (1 − γ )3 − γ 3 ν̄α2 ν̄β2 δ SS + −3γ (1 − γ )2 − 3γ 2 γ ν̄α2 ν̄β2 − γ 3 δν̄ Sα Sβ
(B39)
14
G i 13 = γ Cα Sβ − (1 − γ ) δC S − γ ν̄α2 + γ ν̄α2 Sα Cβ − γ ν̄α2 δ SC
(B40)
14
gi 14 = 2γ (1 − γ ) δCC + 2γ (2γ − 1)(1 − Cα Cβ ) +
+ (1 − γ )2 + γ 2 ν̄α2 ν̄β2 ) δ SS + 2γ γ − 1 + γ ν̄α2 ν̄β2 + γ 2 δν̄ Sα Sβ
24
gi 12
=
(B41)
(B42)
24
gi 12
+ ρωc −2(1 − γ )γ Cα Sβ + (1 − γ )2 δC S +
!
+ ρωc − 2γ γ ν̄α2 + γ 2 ν̄α2 Sα Cβ − γ 2 ν̄α2 δ SC
c
(B43)
24
gi 13 = − ν̄α2 Sα Sβ + ν̄α2 δ SS
(B44)
34
34
2 gi 12
− ρ 2 c2 ω2 {4γ γ (γ − 1)(2γ − 1)(1 − Cα Cβ ) − 2γ 2 (1 − γ )2 δCC }+
gi 12 =
c
−ρ 2 c2 ω2 (1 − γ )4 + γ 4 ν̄α2 ν̄β2 δ SS + −4γ (1 − γ )3 + 4γ 3 γ ν̄α2 ν̄β2 + γ 4 δν̄ Sα Sβ .
(B45)
For the bottom layer matrix T̄ , the partial derivatives of the subdeterminants with respect to phase velocity are:
"
#
β 4 2 l l να νβ − l 2 να νβ + να νβ
8ω2
12
t̄ 12 =
+ 3
4ω4
c
να2 νβ2
12
t̄ 13 =
ν β
(B47)
4 ρ ω2 νβ2
12
β 2 l να νβ − l(να νβ + να νβ )
T̄ 14
12
T̄ 14 = −
+
c
4 ρ ω3 c να2 νβ2
12
t̄ 24 = −
(B46)
(B48)
να
4 ρ ω2 να2
(B49)
2να νβ + c (να νβ + να νβ )
12
t̄ 34 = −
.
4 ρ 2 ω2 c3 να2 νβ2
(B50)
B2 S-wave velocity β
Remembering that γ is also a function of β, the expressions of the partial derivatives with respect to the S-wave velocity of the layer matrix
subdeterminants which are relevant for dispersion function computation are:
12
gi 12 = 2γβ (1 − 2γ )(1 − Cα Cβ ) + (2γ 2 − 2γ + 1)Cα Cββ +
β
(B51)
− 2γβ (γ − 1) + 2γ γβ ν̄α2 ν̄β2 + γ 2 ν̄α2 ν̄β2 β Sα Sβ − (1 − γ )2 + γ 2 ν̄α2 ν̄β2 Sα Sββ
12
gi 13 =
1 Cα Sββ − ν̄α2 Sα Cββ
ρωc
12
12
14
G i 14 = G i 23 = G i 34 =
(B52)
β
1 −2γβ (1 − Cα Cβ ) − (1 − 2γ )Cα Cββ +
ρωc
!
1 2 2
+
−γβ − γβ ν̄α ν̄β − γ ν̄α2 ν̄β2 β Sα Sβ + 1 − γ − γ ν̄α2 ν̄β2 Sα Sββ
ρωc
1 2 12
13
gi 24 = gi 34 =
ν̄β β Cα Sβ + ν̄β2 Cα Sββ − Sα Cββ
β
β
ρωc
β
β
β
(B53)
(B54)
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2007 The Author, GJI, 170, 217–238
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Journal compilation Computation of partial derivatives of Rayleigh-wave phase velocity
12
gi 34 = −
2
1
2 2
2
C
+
1
+
ν̄
ν̄
S
+
ν̄
S
S
−2C
S
ν̄
α
ββ
α
ββ
α
β
α
β
α
β
β
β
ρ 2 ω2 c2
13
gi 12 = ρωc 2γ γβ ν̄β2 Cα Sβ + γ 2 ν̄β2 β Cα Sβ + γ 2 ν̄β2 Cα Sββ +
β
+ ρωc 2γβ (1 − γ )Sα Cβ − (1 − γ )2 Sα Cββ
(B55)
(B56)
13
gi 13 = Cα Cββ
(B57)
β
13 G i 14 = −γβ Sα Cβ + (1 − γ ) Sα Cββ + γβ ν̄β2 Cα Sβ + γ (ν̄β2 )β Cα Sβ + γ ν̄β2 Cα Sββ
β
13
gi 24 = − ν̄β2 β Sα Sβ − ν̄β2 Sα Sββ
(B58)
(B59)
β
235
14 23
G i 12 = G i 12 = ρωc γβ (−6γ 2 + 6γ − 1)(1 − Cα Cβ ) − (γ 2 − γ )(1 − 2γ )Cα Cββ +
β
β
+ ρωc −3(1 − γ )2 γβ − 3γ 2 γβ ν̄α2 ν̄β2 − γ 3 ν̄α2 ν̄β2 β Sα Sβ +
+ ρωc (1 − γ )3 − γ 3 ν̄α2 ν̄β2 Sα Sββ
14 G i 13 = − (1 − γ ) Cα Sββ − γβ Cα Sβ + γβ ν̄α2 Sα Cβ + γ ν̄α2 Sα Cββ
(B60)
(B61)
β
14
gi 14 = 2γβ (2γ − 1)(1 − Cα Cβ ) − 2(γ 2 − γ )Cα Cββ +
β
+ 2(γ − 1)γβ + 2γ γβ ν̄α2 ν̄β2 + γ 2 ν̄α2 ν̄β2 β Sα Sβ + (1 − γ )2 + γ 2 ν̄α2 ν̄β2 Sα Sββ
(B62)
24
gi 12 = ρωc 2(γ − 1)γβ Cα Sβ + (1 − γ )2 Cα Sββ − 2γ γβ ν̄α2 Sα Cβ − γ 2 ν̄α2 Sα Cββ
(B63)
24
gi 13 = −ν̄α2 Sα Sββ
(B64)
34
gi 12 = −ρ 2 c2 ω2 4γ γβ 1 + 2γ 2 − 3γ (1 − Cα Cβ ) − 2γ 2 (1 − γ )2 Cα Cββ +
β
−ρ 2 c2 ω2 (1 − γ )4 + γ 4 ν̄α2 ν̄β2 Sα Sββ +
!
−ρ 2 ω2 c2 −4γβ (1 − γ )3 + 4γ 3 γβ ν̄α2 ν̄β2 + γ 4 ν̄α2 ν̄β2 β Sα Sβ .
(B65)
β
β
The partial derivatives of matrix-T̄ subdeterminants with respect to S-wave velocity of the bottom half-space are:
12
4 t̄ 12
β 4 2 l lβ νβ − l 2 νββ
12
t̄ 12 =
+
β
β
4 ω4 να νβ2
12
t̄ 13 =
β
νββ
4 ρ ω2 νβ2
(B67)
12
2 T̄ 14
β 2 (lβ νβ − l νββ )
12
T̄ 14 =
+
β
β
4 ρ ω3 c να νβ2
(B68)
12
t̄ 24 = 0
(B69)
β
12
t̄ 34 = −
β
(B66)
νββ
.
4 ρ 2 ω2 c2 να νβ2
(B70)
B3 P-wave velocity α
The partial derivatives of the layer matrix subdeterminants with respect to the P-wave velocity of the layer are:
12
= (2γ 2 − 2γ + 1)Cαα Cβ − γ 2 ν̄ 2 ν̄ 2 Sα Sβ − (1 − γ )2 + γ 2 ν̄ 2 ν̄ 2 Sαα Sβ
gi (B71)
12
gi 13 =
(B72)
α α
12 α
α
β
1 Cαα Sβ − ν̄α2 α Sα Cβ − ν̄α2 Sαα Cβ
ρωc
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M. Cercato
236
12 G i 14 =
1 (2γ − 1)Cαα Cβ + 1 − γ − γ ν̄α2 ν̄β2 Sαα Sβ − γ ν̄α2 α ν̄β2 Sα Sβ
ρωc
(B73)
1 2
ν̄β Cαα Sβ − Sαα Cβ
ρωc
(B74)
α
12
gi 24 =
α
12
gi 34 = −
α
1
−2Cαα Cβ + (1 + ν̄α2 ν̄β2 )Sαα Sβ + ν̄α2 α ν̄β2 Sα Sβ
ρ 2 ω2 c2
(B75)
13
gi 12 = ρωc γ 2 ν̄β2 Cαα Sβ − (1 − γ )2 Sαα Cβ
(B76)
13
gi 13 = Cαα Cβ
(B77)
α
α
13 G i 14 = (1 − γ )Sαα Cβ + γ ν̄β2 Cαα Sβ
(B78)
α
13
gi 24 = −ν̄β2 Sαα Sβ
(B79)
α
14 G i 12 = ρωc −γ (γ − 1)(1 − 2γ )Cαα Cβ − γ 3 ν̄α2 α ν̄β2 Sα Sβ +
α
+ρωc (1 − γ )3 − γ 3 ν̄α2 ν̄β2 Sαα Sβ
14
G i 13 = (γ − 1)Cαα Sβ − γ ν̄α2 α Sα Cβ − γ ν̄α2 Sαα Cβ
(B80)
(B81)
α
14 G i 14 = 2γ (1 − γ )Cαα Cβ + (1 − γ )2 + γ 2 ν̄α2 ν̄β2 Sαα Sβ + γ 2 ν̄α2 α ν̄β2 Sα Sβ
(B82)
α
24
gi 12 = ρωc (1 − γ )2 Cαα Sβ − γ 2 ν̄α2 α Sα Cβ − γ 2 ν̄α2 Sαα Cβ
(B83)
24
gi 13 = − ν̄α2 α Sα + ν̄α2 Sαα Sβ
(B84)
34
gi 12 = −ρ 2 c2 ω2 −2γ 2 (1 − γ )2 Cαα Cβ +
α
−ρ 2 c2 ω2 (1 − γ )4 + γ 4 ν̄α2 ν̄β2 Sαα Sβ + γ 4 ν̄α2 α ν̄β2 Sα Sβ .
(B85)
α
α
The partial derivatives of matrix-T̄ subdeterminants with respect to P-wave velocity of the bottom half-space are:
β 4 l 2 ναα
12
t̄ 12 = − 4 2
α
4 ω να νβ
12
t̄ 13 = 0
α
12
T̄ 14 = −
α
12
t̄ 24 = −
α
12
t̄ 34 = −
α
β 2 l ναα
4 ρ ω3 c να2 νβ
(B86)
(B87)
(B88)
ναα
4 ρ ω2 να2
(B89)
ναα
.
4 ρ 2 ω2 c2 να2 νβ
(B90)
B4 Density ρ
The partial derivatives of layer matrix subdeterminants with respect to the density of the layer are:
12
13
13
13
14
14
24
gi 12 = gi 13 = G i 14 = gi 24 = gi 13 = gi 14 = gi 13 = 0
ρ
ρ
ρ
ρ
ρ
ρ
(B91)
ρ
12
gi 13
12
gi 13 = −
ρ
ρ
12
G i 14
12
G i 14 = −
ρ
ρ
(B92)
(B93)
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2007 The Author, GJI, 170, 217–238
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Journal compilation Computation of partial derivatives of Rayleigh-wave phase velocity
12
12 G i 24
G i 24 = −
ρ
ρ
12
2 gi 34
12
gi 34 = −
ρ
ρ
12
gi 13
13
gi 12 =
ρ
ρ
14
G i 12
14
G i 12 =
ρ
ρ
24
gi 12
24
gi 12 =
ρ
ρ
34
2 gi 12
34
gi 12 =
.
ρ
ρ
237
(B94)
(B95)
(B96)
(B97)
(B98)
(B99)
The partial derivatives of matrix-T̄ subdeterminants with respect to P-wave velocity of the bottom half-space are:
12
t̄ 12 = 0
ρ
12
t 13
β2
12
=
t̄ 13 = −
2
ρ
ρ
4 ρ ω2 νβ
12
T 14
β2
l
12
−2
=− 2 3
T̄ 14 = −
ρ
ρ
4 ρ ω c να νβ
12
t
1
12
t̄ 24 = − 24 = − 2 2
ρ
ρ
4 ρ ω να
12
2 t 34
2
ω2
12
t̄ 34 = −
=− 3 4
−1 .
ρ
ρ
4ρ ω
c2 να νβ
(B100)
(B101)
(B102)
(B103)
(B104)
B5 Layer thickness h
If we set:
CC = (Cαh Cβ + Cα Cβh )
(B105)
SS = (Sαh Sβ + Sα Sβh )
(B106)
C S = (Cαh Sβ + Cα Sβh )
(B107)
SC = (Sαh Cβ + Sα Cβh ).
(B108)
The partial derivatives of layer matrix subdeterminants with respect to the thickness of the layer are:
12
gi 12 = 2γ 2 − 2γ + 1 CC − (1 − γ )2 + γ 2 ν̄α2 ν̄β2 SS
(B109)
12
gi 13 =
1 C S − ν̄α2 SC
ρωc
(B110)
12 G i 14 =
1 (2γ − 1)CC + 1 − γ − γ ν̄α2 ν̄β2 SS
ρωc
(B111)
12 G i 24 =
1 2
ν̄ C S − SC
ρωc β
(B112)
h
h
h
h
12
gi 34 = −
h
1
ρ 2 ω2 c2
−2CC + (1 + ν̄α2 ν̄β2 ) SS
13
gi 12 = ρωc γ 2 ν̄β2 C S − (1 − γ )2 SC
h
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(B114)
238
M. Cercato
13
gi 13 = CC
(B115)
h
13 G i 14 = (1 − γ ) SC + γ ν̄β2 C S
(B116)
h
13
gi 24 = −ν̄β2 SS
(B117)
h
14 G i 12 = ρωc γ (1 − γ )(1 − 2γ )CC + (1 − γ )3 − γ 3 ν̄α2 ν̄β2 SS
(B118)
14 G i 13 = (γ − 1)C S − γ ν̄α2 SC
(B119)
14 G i 14 = 2γ (1 − γ )CC + (1 − γ )2 + γ 2 ν̄α2 ν̄β2 SS
(B120)
h
h
h
24
gi 12 = ρωc (1 − γ )2 C S − γ 2 ν̄α2 SC
(B121)
24
gi 13 = −ν̄α2 SS
(B122)
34
gi 12 = −ρ 2 c2 ω2 −2γ 2 (1 − γ )2 CC + (1 − γ )4 + γ 4 ν̄α2 ν̄β2 SS .
(B123)
h
h
h
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2007 The Author, GJI, 170, 217–238
C 2007 RAS
Journal compilation