Finite-frequency tomography

Lapo Boschi (lapo@erdw.ethz.ch)

October 14, 2009

Application of Born theory (scattering theory) to the ray-theory

solution (the forward problem)

In earlier lectures we have seen how the tomographic inverse problem can be formulated onthe basis of ray theory, i.e. the approximation that ω � 1. This approximation leads tothe introduction of the eikonal equation, on whose basis seismic ray paths are defined andtravel-time or phase anomalies are written as integrals of Earth’s structure along the ray-pathonly: in the ray-theory approximation, sensitivity kernels associated to such measurementsare zero everywhere except along the ray path.

The requirement that ω � 1 limits the quality of our modeled seismic waves (whosefrequency, however high, is finite), and therefore the quality of tomographic maps based onray theory. The limit it imposes will be more severe as frequency decreases and we moveaway from a regime of very high frequency.

How can the resolution limit caused by this flaw in the theory be quantified? as frequencydecreases, and the concept of ray path becomes meaningless, what effect will Earth structureaway from the ray path have on the seismograms, on the time and phase anomaly thatwe observe? how should we change our formulation of the inverse problem to account fornon-ray-theoretical phenomena?

One way to answer these questions is to solve numerically the equations of motion of theEarth, with no approximations. But this is very expensive and practically unaffordable.

Another proposed approach involves perturbative theory, or Born theory, that physicistsare already familiar with. To see how Born theory works in global seismology, let us once againgo back to the Earth’s equation of motion, written as a differential equation in displacement,which we dubbed equation (*) in the first lecture of this course. Let L be an operator suchthat eq. (*) can be simply written

Lu = 0. (1)

In earlier lectures we assumed that the effect of the earthquake excitation, necessary to havea nontrivial solution to (1), could be prescribed in the form of an initial/boundary condition.It is now more convenient to write it as a body force equivalent f , replacing (1) with

Lu = f . (2)

If f is impulsive, eq. (2) is called “Green’s problem”; its solution is denoted G and calledGreen’s function1. It is useful to write G as a function of the source and receiver positions,

1The concept of Green’s problem is nicely illustrated, for example, by Dahlen and Tromp, Theoretical

1

which we shall denote rS and rR, respectively; then G = G(rR, rS). The complete Green’sfunction here is a 3 × 3 tensor, and eq. (2) has to be solved for three different impulsiveforcing terms f , each oriented in the direction of one of the axes. It can be proved that, oncethe Green’s problem is solved, the displacement field u associated with any seismic sourcecan be derived, by a simple convolution of a function describing the source in time and space,and the Green’s function G.

We can use the ray-theory approach to solve the Green’s problem and find a Green’stensor G0, valid at high frequencies. Let us then introduce a small perturbation δµ(r), δλ(r),δρ(r), in the parameters describing Earth’s structure. µ, λ and ρ, are replaced by µ + δµ,λ+δλ and ρ+δρ in the analytical expression for L; after linearization, a perturbation operatorδL can be defined such that the perturbed Green’s problem can be written

(L + δL)(G0 + δG) = f , (3)

where everything is known apart from the perturbation δG to the Green’s function. Fromeq. (3), neglecting second order terms,

LG0 + δLG0 + LδG = f , (4)

and since LG0 = f , we are left with

LδG = −δLG0, (5)

where the right hand side is known. Having already solved (in the ray-theory approximation)the Green’s problem associated with the operator L, we are able to find δG by a simpleconvolution of G0 with the new forcing term −δLG0. In the frequency domain,

δG = −∫

VG0(rR,x) · δLG0(x, rS)d3x, (6)

with x denoting the integration variable. The response u of the perturbed Earth model (µ+δµ,etc.) at any location r, to any earthquake with arbitrary hypocenter rS , can now be foundconvolving G(r, rS) + δG(r, rS) with the appropriate source function.

Equation (6) is often interpreted as follows: a wave travels, according to ray theory, fromrS to x. Once it is hit, the point x becomes a secundary source and another wave travels fromx to rR. The cumulative effect of all possible secundary sources is integrated. The importanceof a secundary source (in principle, any point in the Earth can function as secundary source)depends on the properties of the operator δL. If δλ(x) = δµ(x) = δρ(x) = 0, then alsoδLG0(x, rS) = 0. If δLG0(x, rS) 6= 0, we say that the point x acts as a scatterer for theincident wave. Born theory is often referred to as scattering theory.

Born theory holds so long as perturbations δµ, δλ, δρ are small. Then, the perturbedGreen’s tensor G0 + δG provides a higher order of accuracy than the ray-theory one, G0.

Scattering and tomography (the inverse problem)

After replacing δLG0 with its explicit expression, eq. (6) can be rewritten in terms of a“scattering tensor” S,

δG =∫

VG0(rR,x) · S(x) ·G0(x, rS)d3x. (7)

Global Seismology, Princeton Univ. Press 1998, section 4.1.7; a more lengthy treatment is given by Aki and

Richards, Quantitative Seismology, chapters 2 and 4.

After some algebra, tensors SP , SS and Sρ are defined so that the scattering tensor can inturn be written

S = SP

(δvP

vP

)+ SS

(δvS

vS

)+ Sρ

(δρ

ρ

). (8)

Replacing (8) into (7), an expression for δG in terms of relative perturbations to compres-sional and shear velocities and density throughout the Earth is found. This is the mainingredient we need to set up a linear inverse problem. The next step, necessary to derivefrom (7) and (8) an equation relating seismic observable to Earth heterogeneity, is to write aquantity that we can observe directly from a seismogram in terms of δG.

Let Γ(τ) denote the cross-correlation of a reference seismogram u0 and a perturbed seis-mogram u1 (the treatment that follows is going to hold independently for any component ofu as defined above),

Γ(τ) =∫ t2

t1

u0(t− τ)u1(t)dt, (9)

with the time window (t1, t2) chosen to isolate the phase of interest. It is reasonable toestablish that the delay time δT between reference and perturbed phase equals the value ofτ for which Γ(τ) is maximum. After expanding Γ(τ) in a Taylor series around τ = 0, andequating to zero the derivative of the Taylor series (up to second order) with respect to τ ,an expression for the delay time in terms of the perturbed seismogram, and therefore δG, isfound2. Eqs. (7) and (8), with explicit expressions for SP , SS , Sρ, are plugged into the latterexpression, and after some algebra kernels KP , KS and Kρ can be defined such that3

δT =∫

V

[KP

(δvP

vP

)+ KS

(δvS

vS

)Kρ

(δρ

ρ

)]d3x, (10)

and heteroengeities in Earth structure are now directly related to the delay time, i.e. some-thing that we can “pick” from a seismogram. The functions KP (x), KS(x) and Kρ(x) havethe peculiar banana-doughnut shape (with zero value on the infinite-frequency ray path) longdiscussed by Dahlen, Nolet and their co-workers at Princeton, and illustrated here in figures1 and 2. Eq. (10) should be compared with eq. (2) of my lecture notes on body-wave to-mography: they relate the same quantities, but the earlier equation rests on pure ray-theoryapproximation, and the integral there is to be performed only along the ray path; eq. (10)here should represent an improvement to that approximation.

Some authors4 have suggested that the application of finite-frequency tomography, asdescribed here, to global body wave databases helps to enhance model resolution, imagingnarrow features like plumes that have been invisible to traditional tomography (figure 3).However, the significance of such improvements is under debate (figure 4).

Surface wave scattering

The above treatment is naturally applied to the case of body waves, where it is possible todefine travel time, and pick it on a seismogram. Surface waves require a separate formulation.The principles, however, are the same, and the results are similar.

2Dahlen, Hung and Nolet, GJI 2000, vol. 141 page 157, section 4.1.3The following equation is equivalent to eq. (77) of Dahlen and co-authors.4e.g., Montelli et al., Science, 303 pages 338ff, 2004.

Figure 1: Finite-frequency kernels for a uniform background medium. From Hung, Dahlenand Nolet, “Frechet kernels for finite-frequency traveltimes–II. Examples”, Geophysical Jour-nal International, 141, pages 175ff, 2000.

Figure 2: Finite-frequency kernels for a spherically symmetric reference Earth model. FromHung et al., 2000.

Figure 3: P-velocity in the mantle from scattering theory: ascending plumes. From Montelliet al., 2004 (supplementary online material).

Figure 4: P-velocity at several depths in the mantle, derived tomographically from ray theory(left) and the finite-frequency approach (right: model of figure 3). From Montelli et al., 2004.

Figure 5: Finite-frequency kernels relating the phase (measured from the seismogram) andthe phase velocity of 150 s Love waves, for two different source-station geometries. FromBoschi, Geophys. J. Int., 167, pages 238–252, 2006.

The problem can be collapsed to two dimensions only, like in the previous lecture (sur-face wave tomography). A (θ, φ)-dependent (not r-dependent) kernel relating δt(ω) and thecorresponding phase velocity anomaly δc(θ, φ) is found at each frequency ω; let us call itKL(θ, φ;ω) for Love waves, or KR for Rayleigh waves; then eq. (9) from the previous lecture(surface wave tomography) is replaced by

δtL(ω) = −ω

∫ray path

KL(θ, φ;ω)δcL(θ, φ;ω)ds (11)

(and an analogous expression for Rayleigh waves), and the inverse problem is set up exactlyas in earlier lectures.

Looking at KL(θ, φ;ω), KR(θ, φ;ω) we find them to be nonzero over very large portionsof the Earth’s surface, if compared to the body wave kernels, KP , etc. As surface waves arewaves of lower frequency than body waves, it is to be expected that, in their case, limits ofthe high-frequency (ray theory) approximation be more evident, and scattering effects morerelevant.

Alternatively, one can leave explicit the dependence of surface wave phase on Earth struc-ture at varying depth; an expression analogous to equation (17) of lecture 3 follows, with thekernels KSH , etc., now accounting for scattering effects as well.

The former approach has been followed, for example, by R. Snieder and other authorsfrom the Utrecht group5, and likewise in the works of the ETH group illustrated in figures 5and 6. In a more recent article6, the Princeton group have worked out the latter, moregeneral, approach. They show that known 2-D equations for phase velocity maps can bederived as a particular case of the 3-D formulation. Note that their “unperturbed” solution(that I have always called ray-theoretical) is now defined as a linear combination of normalmodes. Hence, it does not involve ray paths. However, the requirement that heterogeneitiesbe smooth is still needed. The rest of the “Princeton” treatment is substantially analogousto what I have outlined in this lecture, though much more detailed.

5Snieder and Nolet, J. geophys. Res. 61 pages 55ff, 1987. Spetzler, Trampert and Snieder, Geophys. J.

Int., 149 pages 755ff, 2002.6Zhou, Dahlen and Nolet, Geophys. J. Int., 158, pages 142ff, 2004.

Figure 6: The 2D inverse problem: phase-velocity maps obtained from inversions based onnumerical finite-frequency (top) and ray theory (bottom), from Rayleigh-wave measurementsat 40 s, 75 s, 150 s and 250 s. From Peter et al., Geophys. Res. Lett., 35, L16315, 2008.Significant differences between ray-theory and finite-frequency results emerge at long periods.