Outline
Comptes Rendus

Internal geophysics (Physics of Earth’s interior)
A Semi-classical calculus of correlations
Comptes Rendus. Géoscience, Volume 343 (2011) no. 8-9, pp. 496-501.

Abstracts

Abstract

The method of passive imaging in seismology has been developped recently in order to image the Earth’s crust from recordings of the seismic noise. This method is founded on the computation of correlations of the seismic noise. In this article, we give an explicit formula for this correlation in the “semi-classical” regime. In order to do that, we define the power spectrum of a random field as the ensemble average of its Wigner measure; this allows phase-space computations: the pseudo-differential calculus and the ray theory. In this way, we get a formula for the correlation of the seismic noise in the semi-classical regime with a source noise which can be localized and non-homogeneous. After that, we show how the use of surface guided waves allows us to image the Earth’s crust.

Résumé

La méthode d’imagerie passive en sismologie a été développée récemment en vue d’imager la croûte terrestre à partir d’enregistrements du bruit sismique. Elle repose sur le calcul des fonctions de corrélation de ce bruit. Nous donnons, dans cet article, des formules explicites pour cette corrélation dans le régime « semi-classique ». Pour cela, nous définissons le spectre de puissance d’un champ aléatoire comme l’espérance de sa mesure de Wigner, ce qui permet d’utiliser un calcul dans l’espace des phases: le calcul pseudo-différentiel et la théorie des « rays ». Nous obtenons ainsi une formule pour la corrélation du bruit sismique dans le régime « semi-classique », avec une source de bruit qui peut être localisée et non homogène. Nous montrons ensuite comment l’utilisation des ondes guidées de surface permet d’imager la croûte terrestre.

Metadata
Received:
Accepted:
Published online:
DOI: 10.1016/j.crte.2011.03.002
Keywords: Passive imaging, Semi-classics, Surface waves, Imagerie passive, Semi-classique, Ondes de surfaces

Yves Colin de Verdière 1

1 Institut Fourier, unité mixte de recherche CNRS-UJF 5582, BP 74, 38402 Saint-Martin-d’Hères cedex, France
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Yves Colin de Verdière. A Semi-classical calculus of correlations. Comptes Rendus. Géoscience, Volume 343 (2011) no. 8-9, pp. 496-501. doi : 10.1016/j.crte.2011.03.002. https://comptes-rendus.academie-sciences.fr/geoscience/articles/10.1016/j.crte.2011.03.002/

Version originale du texte intégral

1 Introduction

Correlation of the noisy wave fields is used as a new tool in seismic imaging and monitoring, starting from the pioneering work of Campillo and Paul (2003) (similar tools have been used in helio-seismology Duvall et al., 1993) and followed by many works (Derode et al., 2004, 2003; Roux et al., 2005; Sabra et al., 2005; Sanchez-Sesma et al., 2006; Shapiro et al., 2005; Shapiro and Campillo, 2004; Weaver, 2005; Weaver and Lobkis, 2004); see also the review paper, (Gouédard et al., 2008). It has also been used in the monitoring of the deformations of volcanoes (Brenguier et al., 2008). Because it is a very powerful method and, hopefully, in order to make it more efficient, it is quite challenging to give mathematical supports to this method, now called “passive imaging”. This has been done in a rather great generality in Colin de Verdière (2009, 2011b) using semi-classical analysis (see also Bardos et al., 2008; Gouédard et al., 2008; Lobkis and Weaver, 2001; Roux et al., 2005).

Exact formulas for the correlations of the fields are known if the source noise is homogeneous (a white noise). This assumption is not satisfied in applications. It is therefore desirable to get formulas valid for more general source noises, in particular if the source noise is localized in some part of the domain. This turns out to be possible in the so-called semi-classical regime where the wave-lengths are negligible with respect to the size of the propagation domain. The field correlation admits a general expression in terms of the Green’s function and the source correlation (Eq. (3)). The idea is to find the asymptotics of this expression in the semi-classical regime.

I will present in this article approximate formulas which are valid in the range of high frequency wave propagation and for which the source noise is localized in some part of the domain of propagation. The correlation is explicitly given in term of the decomposition of the Green’s function as a sum over rays and the (phase-space) power spectrum of the source noise. I can use ray theory if I assume that the source noise has a short correlation distance of the same order of magnitude as the wavelengths. This calculus can be presented in a very geometric way using ray propagation, as well as a re-interpretation of the source correlation in terms of the phase space power spectra. I use the calculus of pseudo-differential operators in a very essential way. I will not reproduce the mathematical arguments which are presented in my paper, Colin de Verdière (2009), but I will try not only to give explicit formulas, but also to present the main ideas and tools.

Here is a more precise description of the content: the goal is to get the formula given in Theorem 5.1 which gives the modification of the correlation of the seismic noise induced by the non-homogeneity of the source noise. The modification is given in terms of the power spectrum of the source noise, the attenuation and the ray dynamics associated to the deterministic wave equation.

I first give a review of the pseudo-differential calculus (Section 2): this allows me to put the basic terminology of rays dynamics and to define power spectra of arbitrary random fields (Section 3).

I then introduce the simplest mathematical model where the source noise is simply the right handside of the wave equation (Section 4) and I present the main formula in Section 5. The interest of the result depends of the relation between the 2 time scales discussed in Section 6: the Ehrenfest time given in terms of the Lyapounov exponent and the attenuation time.

How to use all of this in imaging problems? I do that (Section 7) in the case of seismology using the effective wave equation for the guided surface waves. The final problem turns out to be an inverse spectral problem whose mathematical solution is known.

Finally, I discuss in Section 8 a related issue, namely the calculus of the correlations of plane waves scattered by an obstacle or an inhomogeneity viewed as random waves: the direction of the waves is supposed to be random and uniform. This way, I show that the result of Sanchez-Sesma et al. (2006) is completely general.

2 A short review of the pseudo-differential calculus and Wigner measures

For the mathematics of pseudo-differential operators, see Dimassi and Sjöstrand (1999); Evans and Zworski (2011); Folland (1989); Trèves (1980).

The pseudo-differential operators (ΨDO’s) were introduced in the 1960s by Calderon, Zygmund, Nirenberg, Hörmander and others as a tool in the study of linear partial differential equations with non-constant coefficients. They provide also the geometrical extension of Hamiltonian formalism of classical mechanics to wave mechanics (see Duistermaat, 1996). In applications to physics, it is often called ray theory (see Popov, 2002). The same tools apply to the study of the semi-classical limit of quantum mechanics and to the high frequency limit of wave equations (acoustic, electromagnetic or seismic waves).

There is a small parameter ɛ > 0 in the theory which is the Planck “constant” ℏ in quantum mechanics and the wave length, or more precisely, the dimensionless ratio between the wave length and the size of the propagation domain for wave equations. Most results are only valid in the limit ɛ → 0, but, for simplicity, the reader can think of ɛ as a fixed, small enough, number.

2.1 ΨDO’s

A pseudo-differential operator (ΨDO) on Rd is a linear operator on functions f:RdC, Aɛ ≔ Opɛ(a), defined using a suitable function defined on the phase space, a(x,ξ):RxdRξdC, called the symbol of Aɛ, by the formula (Weyl quantization)

Aɛ(f)(x)=1(2π)dRd×Rdei(xy).ξax+y2,ɛξf(y)dydξ.

The function a is assumed to be smooth and homogeneous near infinity in ξ. The Schwartz kernel1 [Aɛ](x, y) of Aɛ is located near the diagonal x = y and is of the form

[Aɛ](x,y)ɛ0k(x,(xy)/ɛ)
where k(x, z) is a smooth function outside z = 0 going to 0 as z → ∞.

Simple examples are:

  • • Opɛ(1) = Id by the Fourier inversion formula;
  • Opɛ(ξj)=ɛixj;
  • • Opɛ(xj) is the multiplication by xj;
  • • If χ is a positive function with bounded support, the operator Opɛ(χ(ξ)) is a frequency filter;
  • • Op(| ξ | 2 + V(x)) = − ℏ2Δ + V(x): the Schrödinger operator;
  • Opɛ(n(x)|ξ|2)=ɛ2divn(x)grad: the acoustic wave operator.

The main properties are the following ones which hold as ɛ → 0:

  • • Composition:
    Opɛ(a)Opɛ(b)Opɛ(ab);
  • • Brackets:
    [Opɛ(a),Opɛ(b)]ɛiOpɛ{a,b}
    where
    {a,b}=j=1daξjbxjaxjbξj
    is the Poisson bracket. This last property is very important because it relates the algebra of ΨDO’s to the geometry of the phase space given by the Poisson bracket.

2.2 Wigner functions

Wigner functions define the localization of energy in the phase space Rxd×Rξd for a wave function u = u(x). They involve the scale ɛ. The Wigner function Wuɛ(x,ξ) of u is the function on the phase space defined by the identities

aC0R2d,R2da(x,ξ)Wuɛ(x,ξ)dxdξ=Opɛ(a)u|u,
where u|v=u(x)v¯(x)dx, or
Wuɛ(x,ξ)=1(2π)dRdeiv.ξux+ɛv2ūxɛv2dv.
I have
RdWuɛ(x,ξ)dξ=|u(x)|2,RdWuɛ(x,ξ)dx=|Fɛu(ξ)|2,
where Fɛu(ξ)= is the ɛ − Fourier transform of u given by
Fɛu(ξ)=1(2πɛ)d/2eix.ξ/ɛu(x)dx.
This means that the marginals of the Wigner measure Wuɛ(x,ξ)dxdξ are |u(x) | 2 dx and |Fɛu(ξ)|2dξ.

2.3 Hamiltonian dynamics and ray method

Let us consider the wave equation uttLu=0 where L is an elliptic ΨDO like the acoustic operator L=div(ngrad). The symbol, usually called the dispersion relation, of this equation is ω2 − n(x)||ξ||2 = 0. To this relation is associated a dynamics called the ray dynamics given by the Hamilton equations:

dxjdt=Hξj,dξjdt=Hxj(1)
with H=nξ. The main result (Theorem 5.1 below) uses the “Hamiltonian flow” Φt: Φt(x, ξ) is the value at time t of the previous differential system (1) with data (x, ξ) at the time t = 0. In the case of an homogeneous medium, n = n0 =constant, I have
Φt(x,ξ)=(x+tn0ξ/ξ,ξ).
In the ray theory, this correspond to the group velocity of waves n0.

The mathematical theory of rays is called the theory of Fourier Integral Operators and has been developed in the 1970s by Hörmander and Duistermaat (Duistermaat, 1996), following some pioneering work of Lax and Maslov. A presentation more adapted to physicists is given in Popov (2002). Unfortunately, the geometrical background is rather sophisticated and cannot be presented in a few pages. However, explicit formulas, in terms of oscillatory functions and oscillatory integrals, are available.

In what follows, I will use the fact that the Green’s function G(t, x, y) of the wave equation admits, in the semi-classical regime (short wave-length), a decomposition as a sum of contributions of rays γ going from y to x in time t: G = ∑ γGγ.

3 Random fields: power spectra and correlations

Let f=f(x),xRd, be a random complex-valued field with zero mean value. Let us denote by E the expectation or ensemble average.

Definition 3.1

The correlation of the random field f is the 2-points function given by

C ( x , y ) E ( f ( x ) f ¯ ( y ) )
The power spectrum of the random field f is the function on the phase space given by the expectation of the Wigner functions
p ɛ E ( W f ɛ ) .

The power spectrum and the correlation contain the same information:

  • • The correlation C(x, y) is ((2πɛ)d times) the operator kernel of Opɛ(p) or
    C(x,y)=eixy|ξ/ɛpɛx+y2,ξdξ.
  • pɛ is ((2πɛ)d times) the symbol of the operator whose integral kernel is C.

Example 3.1

White noiseC = δ(x − y), pɛ = 1/(2πɛ)d.

Example 3.2

Stationary noise on R with ɛ = 1, C(s, t) = F(s − t) and p1(s, ω) is the Fourier transform F(F)(ω).

4 A mathematical model

I will now discuss the basic mathematical model: it consists of 2 parts:

  • • A deterministic wave equation which could be the elastic wave equation or more simply here the acoustic wave equation. Because the source of noise will be permanent, some attenuation in the equation is needed.
  • • A source noise assumed to be stationary and ergodic in time. In seismology, this source is usually created by the interaction of the fluids surrounding the earth crust (atmosphere or ocean) with the crust itself. This source is modeled by a random field which I put on the right-handside of the equation.

For simplicity, I will discuss only the case of a scalar acoustic wave equation on some domain in Rxd with a random source field f = f(x, t) (t is the time):

utt+a(x)utLu=f(2)
where
  • • The field u = u(x, t) is scalar;
  • a, the attenuation, is a smooth >0 function. I will assume for simplicity that a is time independent, but it is not really necessary;
  • L is a self-adjoint pseudo-differential operator of symbol ɛ2l02(x,ξ). Usually, l0 is homogeneous of degree 1 which makes L independent of ɛ. This will not be the case for dispersive waves like surface waves. Typical examples are the Laplace-Beltrami operator of a Riemannian metric on X with l0(x,ξ)=gij(x)ξiξj and the acoustic wave operator divn(x)grad) with l0(x,ξ)=n(x)ξ. I introduce L0Opɛ(l0)=ɛL;
  • f = f(x, t) is a stationary and ergodic (in time) random field with correlation E(f(s,x)f(s,y))=δ(ss)Γ(x,y) and power spectrum p(x, ξ); I assume that p(x, ξ) has bounded support and that f is real valued and hence that p(x, ξ) is even w.r. to ξ. I assume that p is independent of ɛ, this implies that the correlation is ɛ − dependent: in particular, Γ(x, y) ≪ | x − y |/ɛ. The source noise decorrelates rapidly as | x − y |≫ɛ.

The Green’s function is the integral kernel G giving the causal solution of Eq. (2) in terms of f:

u(x,t)=0dsXG(s,x,y)f(ts,y)dy.
Our goal is to compute the correlation
CA,B(τ)=limT+1T0Tu(A,t)u(B,tτ)dt.

Lemma 4.1

The following relation holds:CA,B(− τ) = CB,A(τ).

Hence I can (and will!) restrict ourselves to τ > 0.

Using the fact that the source noise is ergodic and stationary, I get the following result:

Theorem 4.1

The field correlation is given by Equation (3) only in terms of the Green’s function G and the correlation Γ of the source noise

CA,B(τ)=0dsX×XdxdyG(s+τ,A,x)G(s,B,y)Γ(x,y).(3)

All the work is now concentrated to get a more explicit and more geometric expression: this will be done using an expression of the Green’s function as a sum over rays going from B to A in time τ and using the power spectrum p of f which is a semi-classical expression of the correlation of the source noise.

5 The main formula

Let us denote by Ω±(t) the “one-parameter groups” of linear operators generated by ±iL0 − ɛa/2: Ω+(t)u0 is the solution of the differential equation u˙=(iɛL0a/2)u with u(0) = u0, and similarly for Ω(t). The use of Ω±(t) is a way to split the Green function of the wave equation usually given by some “sinus” function into 2 exponentials: this way, I reduce the wave equation from an equation with of second order in time to a diagonal system of first order in time.

Ω±(t)=et±iɛL0a/2=et±iLa/2.
I will express the result in terms of operators instead of expressing them in terms of their kernels (matrices). This gives a much more compact expression! The symbol ∘ means the composition of operators while Ĉ(τ) is the operator whose integral kernel (matrix) is CA,B(τ):
(Ĉ(τ)u)(A)=XCA,B(τ)u(B)dB.
The main result is:

Theorem 5.1

The correlation is given, forτ > 0, as ɛ goes to 0, by

Ĉ(τ)Ω+(τ)+Ω(τ)Π,(4)
with Π=Opɛ(π¯) and
π¯(x,ξ)=ɛ24l020et0a(Φs(x,ξ))dsp(Φt(x,ξ))dt,(5)
and if a = a0 is constant
π¯(x,ξ)=ɛ24l020ea0|t|p(Φt(x,ξ))dt.

I will compare our result (Eqs. (4) and (5)) to the Green’s function.

In the semi-classical regime, i.e. as ɛ → 0, I have

G(t,A,B)ɛ2i(Ω+(t)Ω(t))L01(A,B),
Let us now compute the τ − derivative of CA,B(τ):
ddτC(τ)ɛiΩ+(τ)Ω(τ)L0Π.
In the case of white noise and constant attenuation a0, I know (see, for example, Colin de Verdière (2009) Section 5.1 for a derivation) that
ddτC(τ)=12a0G(τ)(6)
which is consistent with the previous semi-classical formula.

I can now give a more concrete formula:

Corollary 5.1

WritingG(τ, A, B) as a sumγGγ of contributions of rays γ(s) with γ(0) = (B, ξB) and γ(τ) = Φτ(B, ξB) = (A, ξA), I get

ddτC(τ,A,B)γMγGγ,
with
Mγ=120et0a(γ(s))dsp(γ(t))dt.

In the case of the white noise p = 1 and a = a0, I recover the formula

Mγ=1/2a0.(7)

Let us also remark that, if there is an unique trajectory from B to A in time τ, the prefactor Mγ applies to the Green’s function itself. It is the case, if I work with wave equations with constant coefficients in Rn.

The previous formula is consistent with the observations of the paper by Stehly et al. (2006): the correlation CA,B(τ) is not always an even function of τ as it is if the source is a white noise. The evenness is valid only up to scaling of CA,B(τ):

CB,A(τ)=CA,B(τ)kCA,B(τ).
The factor k is the ratio of the integrals giving Mγ for the ray γ(t) going from B to A and γ(− t) going from A to B.

6 Time scales

As I see from the general expression of the correlation given in Eq. (3), the proof of the main theorem 5.1 involves the knowledge of the Green’s function at large times. This is a well known difficulty and the semi-classical expansions of the Green’s functions are valid up to the so-called Lyapounov time which involves the Lyapounov exponent measuring the rate of instability of the ray dynamics. Roughly speaking, the Lyapounov exponent is the smallest number λ so that the distance between any to rays γ1(t) and γ2(t) satisfies the estimates

d(γ1(t),γ2(t))Ceλtd(γ1(0),γ2(0)
with C independent of γ1(0) and γ2(0). There is an associated time scale TLyap = 1/λ. On the other hand there is an attenuation time scale for the wave dynamics expressed in terms of the decay of the Green’s function
|G(t,x,y)|CeT/Tatt.
Tatt satisfies the estimate Tatt ≥ 2/inf  a. The approximation given in Theorem 5.1 is better when Tatt ≫ TLyap. In particular, this condition is necessary in order to get point-wise convergence (i.e. convergence for A and B fixed).

7 The use of surface waves for passive imaging

A remarkable application of the previous tool is to the imaging of the Earth’s crust (Campillo and Paul, 2003; Lobkis and Weaver, 2001; Shapiro and Campillo, 2004; Shapiro et al., 2005; Weaver and Lobkis, 2001, 2002). This is done using the part of the Green’s function associated to the surface waves: the earth crust acts as a wave guide on elastic waves and these waves follow an effective wave equation. The effective Hamiltonian is described now: let us start with the acoustic wave equation utt − div(n gradu) = 0 with the function n coming from a stratified medium n = n(x, z) (here z = 0 is the surface) where n is weakly dependent of x (this can be formalized as n(x, z) = N(ɛx, z) with N smooth and ɛ small). Using the adiabatic separation of variables u ∼ U(ɛx, z)eix|ξ with U weakly dependent of x, I can operate as if n was independent of x and I get the reduced equation

Utt+Op1(λ(x,ξ))U=0
where λ(x, ξ) is an eigenvalue of the Sturm-Liouville operator
Lx,ξ=ddzn(x,z)ddz+n(x,z)ξ2
with appropriate boundary conditions at z = 0.

From the correlation, I get the ray dynamics of the surface waves and hence the effective Hamiltonians λ(x, ξ). The inverse problem to be solved is the following inverse spectral problem: from the fundamental mode (or any other available mode) of Lx0,ξ in some range of wave numbers |ξ |, recover n(x0, z). This is the kind of well posed inverse problem for which analytical/numerical method can be used (see Colin de Verdière, 2011a).

8 A formula for the scattering of random plane waves

I have seen an exact formula for the correlation of the wave field when the attenuation a is constant and the source noise is a white noise. I will see another exact formula in the context of wave scattering by a perturbation sitting in a bounded domain of Rd (see Colin de Verdière, 2011b). This formula is very general and applies in all situations of wave scattering (scalar or elastic waves), i.e. for any medium which is homogeneous near infinity: non-homogeneity’s lies at finite distances or there is a scattering by a bounded obstacle. This calculus was motivated by the result of Sanchez-Sesma et al. (2006), showing that this result is completely general.

Let us consider for example an acoustic wave equation (2) with n = n0 outside a bounded set of Rd. I will consider scattering solutions of the stationary wave equation

div(ngradu)ω2u=0(8)
which are of the following form: let us define, for kRd, the plane wave
e0(x,k)=eik.x.
I am looking for solutions
e(x,k)=e0(x,k)+es(x,k)
of Eq. (8) in Rd, with n0k2 = ω2 (2), where es, the scattered wave, satisfies the so-called Sommerfeld radiation condition:
es(x,k)=eik|x||x|(d1)/2ex|x|,k+O1|x|,x.
The complex function e(xˆ,k) is usually called the scattering amplitude and is a signature of the inhomogeneities. The functions e(x, k) are deformed exponentials and allow to write an explicit spectral decomposition of our wave operator, which is a “deformation” of the Fourier transform.

Let us look at e(x, k) as a random wave with k=ω/n0 fixed. The point-point correlation of such a random wave Cωscatt(x,y) is given by:

Cωscatt(x,y)=kn0=ωe(x,k)e(y,k)¯dσ(kˆ).
It is proved in Colin de Verdière (2011b), section 8, that
Cωscatt(x,y)=2d+1πd1n0d/2ωd2(G(ω+i0,x,y)),
where G(ω, x, y) is the stationary Green’s function, i.e. the Schwartz kernel of ω2+div(ngrad)1.

9 Conclusions

I hope to have convinced the reader, even if he is not very much involved in mathematics, that it is possible to derive rather explicit asymptotic formulas for the correlation CA,B(τ) of seismic noise. The main conclusion is that, in the semi-classical regime, even if the source noise is not homogeneous, the field correlation is very close to the Green’s function; in many cases, there is only a prefactor which I computed and which introduces no phase shift. This prefactor vanishes if the support of the source noise does not meet the rays from B to A.

Many other ideas and applications remains to be exploited:

  • • Is it possible to use the previous tools in order to get informations on the source noise?
  • • Can I extend the previous calculus to the case where the source noise is located on a surface?
  • • Can I do something similarly in other regimes of propagation, in particular in non-smooth media?
  • • Can I get applications of the general formula to monitoring?

1 The “Schwartz kernel” of a linear operator A is the “continuous matrix” of A, we will denote it by [A](x, y) and it is characterized by Af(x) = ∫ X[A](x, y)f(y) dy.

2 As often, I denote k ≔ | k | and kˆk/k.


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