I042
Minimum-phase Property of Memory Functions in
the Wave Equation
K.N. van Dalen* (Delft University of Technology), E.C. Slob (Delft University of Technology) & F.C. Schoemaker (Delft University of Technology)
SUMMARY
Memory functions occur in the wave equation as time-convolution operators and generally account for the instantaneous and non-instantaneous responses of a medium. The specific memory function that is causal and stable, and the inverse of which is also causal and stable, is conventionally referred to as minimum phase. In this paper we present "extended minimum-phase relations" between the amplitude and phase spectra of a memory function that has different properties. The considered memory function and its inverse are both causal, but they do not need to be stable. We still address the function as minimum phase because the phase spectrum exhibits minimum group delay, like a conventional minimum-phase function. We have successfully tested the derived relations for the well-known Maxwell and Kelvin-Voigt models. The relations have potential applications in acoustics, seismology, poroelasticity, electromagnetics, electrokinetics and any other effective-medium theory that employs memory functions.
Introduction
Memory functions show up as time-convolution operators in the governing equations for wave propaga-tion. For instance, in the equations for a homogeneous dissipative fluid [outside the source domain (de Hoop, 1995)],
∂2
t (ρ∗ u) +∇p= 0, κ∗ p +∇· u = 0, (1) ρ=ρ(t) andκ =κ(t) are density and compressibility memory functions, respectively, that generally
account for both the instantaneous and non-instantaneous responses of the medium. In Eq. (1) u is the particle-displacement vector, p the acoustic pressure and the asterisk represents temporal convolu-tion. Probably the most well-known models for memory functions are the rheological models based on springs and dashpots that are particularly used to describe creep and relaxation phenomena (e.g., Bour-bié et al., 1987). Other examples are the frequency-dependent viscodynamic function and the dynamic permeability in poroelasticity, and memory functions that account for energy loss in scattering media. One of the most important properties of a memory function f(t) is its causal character, i.e., f (t) = 0
for t< 0. The fact that f (t) is a causal function of time implies that the real and imaginary parts of
the spectrum ˆf(ω) (ωdenotes angular frequency) are related via the well-known Kramers-Kronig (KK) relations (Toll, 1956). A specific memory function that is also stable (absolutely integrable) and the inverse of which is causal and stable as well, possesses the minimum-phase property (Oppenheim and Schafer, 1989). This implies that the phase spectrum has minimum group delay, which means that the phase is the smallest possible for a given amplitude spectrum such that f(t) remains causal (Aki
and Richards, 2002). It also implies that another pair of relations hold that arise from the fact that the Laplace-transformed function ˆf(s) (s is Laplace parameter) is free of zeros and singularities in the right
half of the s-plane (RHP). These “minimum-phase (MP) relations” between the amplitude and phase spectra provide a big advantage for the experimental determination of a memory function: either the amplitude or the phase needs to be measured.
When both a function and its inverse are causal and stable, it must consist of an instantaneous part (described by the Dirac delta function) and a delay part. In many cases the specific memory function and/or its inverse are/is not stable. For instance, the function might include a Heaviside step function. For such a function the MP relations do not hold. However, as both the function and its inverse are causal functions of time, it is still true that the RHP is free of zeros and singularities, and therefore the phase spectrum should still have minimum group delay (Aki and Richards, 2002). Hence, it is challenging to see whether any MP relations can be derived between the amplitude and phase spectra. In this paper we present first results using scaling of the spectra. First, we briefly review the conventional MP relations. Then, we summarize the derivation of extended MP relations and discuss their applicability. We test the relations for the well-known Maxwell and Kelvin-Voigt models and end up with conclusions.
Conventional minimum-phase relations
We summarize the derivation of the MP relations for a real-valued function f(t) that is both causal and
stable. The derivation runs parallel to that of the original KK relations between the real and imaginary parts of the Laplace transform ˆf(s) of a causal function (de Hoop, 1995). As the function f (t) and its
inverse are both causal, ˆf(s) has no singularities in the RHP and, in addition, no zeros in that part of
the s-plane. As a result, ln( ˆf(s)) is regular in the RHP. Further, as f (t) is stable, the behaviour of | ˆf(s)| is o(s) as |s| →∞in the RHP, i.e., lim|s|→∞| ˆf(s)/s| = 0. This implies that lim|s|→∞fˆ(s) = f∞is a real constant; this constant corresponds to the instantaneous (Dirac) part of f(t).
To derive MP relations between the amplitude and phase of ˆf(s), we consider the ratio (ln( ˆf(s)) −
ln( f∞))/(s − iΩ), whereΩis real. In the numerator we take ln( ˆf(s)) instead of simply ˆf(s) like in the
derivation of the KK relations (de Hoop, 1995); the subtraction in the numerator is referred to below. We choose an integration contour C in the s-plane consisting of the imaginary axis, a semicircle C∞of
( ) Res ( ) Im s i s = Ω C∞ ( ) Re s ( ) Im s C i s = Ω C∞ 0 C 0 s = Ω CΩ a) b)
Figure 1 Closed contour of integration in the complex s-plane used to establish the (a) conventional and (b) the extended minimum-phase relations. The direction of integration is indicated.
infinite radius and a semicircle CΩof infinitesimal radius around s= iΩ, both in the RHP; see Figure 1a. For this contour, by virtue of Cauchy’s theorem it holds that
˛
C
ln( ˆf(s)) − ln( f∞)
s− iΩ ds= 0. (2)
The contribution of C∞vanishes due to the subtraction in the numerator because|ln( ˆf(s)) − ln( f∞)| =
o(1) as |s| →∞in the RHP (de Hoop, 1975). This enables us to establish a relation between the contribu-tions along the imaginary axis and around s= iΩ; taking them together according to Eq. (2), separating between the real and imaginary parts using ln( ˆf(s)) = ln | ˆf(s)| + iϕ(s) and putting s = iω, we obtain
ln| ˆf(Ω)/ f∞| = −1 π− ˆ ∞ −∞ ϕ(ω) −ϕ∞ ω−Ω dω, (3) ϕ(Ω) −ϕ∞= +1 π− ˆ ∞ −∞ ln| ˆf(ω)/ f∞| ω−Ω dω. (4)
These are the conventional MP relations between the amplitude and phase spectra; they form a Hilbert-transform pair (−´ is the principal-value integral) and are very similar to the KK relations (de Hoop, 1995).
Extended minimum-phase relations
In many cases the memory function f(t) does not obey the specific requirement of | ˆf(s)| being o(s) as
|s| →∞in the RHP. When an instantaneous part is absent ˆf(s) goes to zero and when time derivatives
of the Dirac function are involved it goes to infinity; in both cases ln( ˆf(s)) becomes unbounded, which
is problematic for the derivation of useful MP relations. However, for any polynomial function with finite powers of s the behaviour of|ln( ˆf(s))| = o(s) as |s| →∞in the RHP, which means that scaling of ln( ˆf(s)) by s solves the problem of unboundedness at infinity because |ln( ˆf(s))/s| = o(1) as |s| →∞in the RHP. For this reason, as a starting point we consider [cf. Eq. (2)]
˛
C
ln( ˆf(s))
s(s − iΩ)ds= 0, (5)
where C now also includes an infinitesimal semicircle C0around the pole at s= 0 (see Figure 1b). Due to the scaling the contribution of C∞vanishes. Taking the other contributions together, separating again
10 20 30 40 50 60 70 80 90 −2 −1.5 −1 −0.5 0 f (Hz) Zener Maxwell Kelvin-Voigt (f) ϕ
Figure 2 Phase spectra of the Maxwell, Kelvin-Voigt and Zener models: analytical (solid black line; see Table 1) and computed using MP relations (dashed gray lines). Here, f =Ω/(2π) denotes frequency.
For K∞,κ∞= 1/K∞,η,τ1andτ2the same parameter values have been used as in Carcione (2007).
between the real and imaginary parts and putting s= iω, we obtain ln| ˆf(Ω)/ ˆf(0)| Ω = − 1 π− ˆ ∞ −∞ ϕ(ω) ω(ω−Ω)dω, (6) ϕ(Ω) −ϕ(0) Ω = + 1 π− ˆ ∞ −∞ ln| ˆf(ω)| ω(ω−Ω)dω, (7)
where the principal-value integral is taken at bothω=Ωandω = 0. We address Eqs. (6) and (7) as
“extended MP relations”. To calculate the amplitude spectrum from the phase spectrum, one obviously needs ˆf(0), which is unknown beforehand. However, in many cases a time-domain signal of f (t) is
known (e.g., from an experiment); then, ˆf(0) can be found by taking the Laplace transform for s = 0, i.e.,
ˆ
f(0) =´∞
0 f(t) dt. The phase spectrum can be calculated straightforward from the amplitude spectrum asϕ(0) = 0 [by virtue ofϕ(s) = −ϕ(s∗) for real-valued f (t)].
It can be shown that Eqs. (6) and (7) are still valid when singularities (poles or zeros) are present in ˆf(s)
on the imaginary s axis at s= ±iΩs [withΩsreal; singularities always appear in pairs for real-valued
f(t), except whenΩs= 0]. In the specific case thatΩs= 0, however, Eq. (6) gives infinity for allΩand
can therefore not be used [because| ˆf(0)| is either 0 or∞], but Eq. (7) remains useful. Further, both the extended MP relations are not valid whenΩ=Ωs= 0.
The derivation here shows that MP relations can be obtained not only for a causal and stable memory function that has a causal and stable inverse, but for any causal memory function [for which ˆf(s) can be
written as a polynomial in s when|s| →∞] that has a causal inverse. This confirms our expectation: a function without zeros and singularities in the RHP should possess the minimum-phase property. Examples
Specific expressions of the memory functions used in three different models (Maxwell, Kelvin-Voigt, Zener) are displayed in Table 1 (Bourbié et al., 1987; Carcione, 2007). These models are associated with creep and relaxation phenomena and are therefore particularly related toκ(t) [see Eq. (1)]. The inverse
ofκ(t) [i.e., the compression-modulus memory function K(t)], and its amplitude and phase spectra are
given in the table as well. We now apply the extended MP relations to ˆκ(ω) of the Maxwell and
Kelvin-Voigt models. The conventional MP relations are applied to ˆκ(ω) of the Zener model [becauseκ(t)
Table 1 The compressibility functionκ(t), its inverse K(t) (the compression-modulus function), and the
amplitude| ˆκ(ω)| and phaseϕ(ω) spectra of the (1) Maxwell, (2) Kelvin-Voigt and (3) Zener models.
The spectra have been obtained after Laplace transformation and subsequently using s= iω;δ(t) is the
Dirac function, ˙δ(t) its time derivative and H(t) the Heaviside function. Further,ηdenotes the dynamic fluid viscosity,κ∞and K∞are the instantaneous parts ofκ(t) and K(t) (if present), respectively, andτ1
andτ2are time scales withτ1>τ2(Carcione, 2007). Note that K∞6= lim|s|→∞Kˆ(s) (in RHP) for model
(2). This limit is unbounded; yet the constant K∞can be distinguished.
κ(t) K(t) | ˆκ(ω)| −ϕ(ω) 1 κ∞δ(t) +η1H(t) κ1∞δ(t) −κ12 ∞ηe − t κ∞ηH(t) qκ2 ∞+η21ω2 arctan 1 κ∞ηω 2 η1e−Kη∞tH(t) K∞δ(t) +ηδ˙(t) √ 1 K2 ∞+η2ω2 arctan ηω K∞ 3 K1∞ δ(t) +τ1τ1τ2−τ2e− t τ1H(t) K∞ δ(t) −τ1τ1τ2−τ2e− t τ2H(t) K∞τ2τ1 r 1+ω2τ2 1 1+ω2τ2 2 arctan ω (τ1−τ2) 1+ω2τ1τ2
from the amplitude spectrum. To this end, we rewrite Eqs. (4) and (7) using symmetry properties and standard integrals (de Hoop, 1995), leading to exactly the same integrals in both cases, i.e.,
2Ω π ˆ ∞ 0 ln| ˆf(ω)/ ˆf(Ω)| ω2−Ω2 dω= ϕ(Ω) −ϕ∞, Eq. (4), ϕ(Ω), Eq. (7). (8)
Here, it should be noted that the properties of the integrands are still different (especially the behaviour of ln| ˆf(ω)| atω →∞). The integrals are now proper ones and the integrands are integrable (also at ω=Ω), which allows straightforward numerical evaluation.
We have evaluated Eq. (8) for the three different models. The results are shown in Figure 2, together with the analytical phase spectra. For each of the models both results coincide, which confirms the validity the applied MP relations. It can be verified that the amplitude spectrum can also successfully be computed from the phase spectra using Eqs. (3) and (6) for the Zener and Kelvin-Voigt models, respectively. For the Maxwell model there is a singularity at s= 0 and hence Eq. (3) cannot be used.
Conclusions
We have derived “extended minimum-phase relations” between the amplitude and phase spectra of a memory function. The considered memory function and its inverse are both causal, but they do not need to be stable like for a conventional minimum-phase function. We address the function as (ex-tended) “minimum-phase” because the phase spectrum still exhibits minimum group delay. We have successfully tested the derived relations for the Maxwell and Kelvin-Voigt models. The relations have potential applications in acoustics, seismology, poroelasticity, electromagnetics, electrokinetics and any other effective-medium theory that employs memory functions.
Acknowledgements
We thank prof. A.T. de Hoop for very interesting discussions. This research is sponsored by The Netherlands Research Centre for Integrated Solid Earth Sciences (ISES).
References
Aki, K. and Richards, P.G. [2002] Quantitative Seismology. Freeman and Co.
Bourbié, T., Coussy, O. and Zinszner, B. [1987] Acoustics of Porous Media. Gulf Publishing.
Carcione, J.M. [2007] Wave Fields in Real Media. Theory and numerical simulation of wave propagation in anisotropic, anelastic, porous and electromagnetic media (2nd edition). Elsevier.
de Hoop, A.T. [1975] Theorie van het elektromagnetische veld (in Dutch). Delftse Universitaire Pers. de Hoop, A.T. [1995] Handbook of Radiation and Scattering of Waves. Academic Press.
Oppenheim, A.V. and Schafer, R.W. [1989] Discrete-Time Signal Processing. Prentice-Hall.
