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 on the basis of ray theory, i.e. the approximation that ω 1. This approximation leads to the introduction of the eikonal equation, on whose basis seismic ray paths are defined and travel-time or phase anomalies are written as integrals of Earth s structure along the ray-path only: in the ray-theory approximation, sensitivity kernels associated to such measurements are zero everywhere except along the ray path. The requirement that ω 1 limits the quality of our modeled seismic waves (whose frequency, however high, is finite), and therefore the quality of tomographic maps based on ray theory. The limit it imposes will be more severe as frequency decreases and we move away from a regime of very high frequency. How can the resolution limit caused by this flaw in the theory be quantified? as frequency decreases, and the concept of ray path becomes meaningless, what effect will Earth structure away from the ray path have on the seismograms, on the time and phase anomaly that we observe? how should we change our formulation of the inverse problem to account for non-ray-theoretical phenomena? One way to answer these questions is to solve numerically the equations of motion of the Earth, with no approximations. But this is very expensive and practically unaffordable. Another proposed approach involves perturbative theory, or Born theory, that physicists are already familiar with. To see how Born theory works in global seismology, let us once again go 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 such that eq. (*) can be simply written Lu = 0. (1) In earlier lectures we assumed that the effect of the earthquake excitation, necessary to have a 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 called Green s function 1. It is useful to write G as a function of the source and receiver positions, 1 The concept of Green s problem is nicely illustrated, for example, by Dahlen and Tromp, Theoretical 1
which we shall denote r S and r R, respectively; then G = G(r R, r S ). The complete Green s function here is a 3 3 tensor, and eq. (2) has to be solved for three different impulsive forcing terms f, each oriented in the direction of one of the axes. It can be proved that, once the Green s problem is solved, the displacement field u associated with any seismic source can 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 s tensor G 0, 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 operator δl can be defined such that the perturbed Green s problem can be written (L + δl)(g 0 + δg) = f, (3) where everything is known apart from the perturbation δg to the Green s function. From eq. (3), neglecting second order terms, LG 0 + δlg 0 + LδG = f, (4) and since LG 0 = f, we are left with LδG = δlg 0, (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 simple convolution of G 0 with the new forcing term δlg 0. In the frequency domain, δg = G 0 (r R, x) δlg 0 (x, r S )d 3 x, (6) V 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 r S, can now be found convolving G(r, r S ) + δg(r, r S ) with the appropriate source function. Equation (6) is often interpreted as follows: a wave travels, according to ray theory, from r S to x. Once it is hit, the point x becomes a secundary source and another wave travels from x to r R. The cumulative effect of all possible secundary sources is integrated. The importance of 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 also δlg 0 (x, r S ) = 0. If δlg 0 (x, r S ) 0, we say that the point x acts as a scatterer for the incident wave. Born theory is often referred to as scattering theory. Born theory holds so long as perturbations δµ, δλ, δρ are small. Then, the perturbed Green s tensor G 0 + δg provides a higher order of accuracy than the ray-theory one, G 0. Scattering and tomography (the inverse problem) After replacing δlg 0 with its explicit expression, eq. (6) can be rewritten in terms of a scattering tensor S, δg = V G 0 (r R, x) S(x) G 0 (x, r S )d 3 x. (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 S P, S S and S ρ are defined so that the scattering tensor can in turn be written S = S P ( δvp v P ) ( ) ( ) δvs δρ + S S + S ρ. (8) v S ρ Replacing (8) into (7), an expression for δg in terms of relative perturbations to compressional and shear velocities and density throughout the Earth is found. This is the main ingredient we need to set up a linear inverse problem. The next step, necessary to derive from (7) and (8) an equation relating seismic observable to Earth heterogeneity, is to write a quantity that we can observe directly from a seismogram in terms of δg. Let Γ(τ) denote the cross-correlation of a reference seismogram u 0 and a perturbed seismogram u 1 (the treatment that follows is going to hold independently for any component of u as defined above), Γ(τ) = t2 t 1 u 0 (t τ)u 1 (t)dt, (9) with the time window (t 1, t 2 ) chosen to isolate the phase of interest. It is reasonable to establish that the delay time δt between reference and perturbed phase equals the value of τ for which Γ(τ) is maximum. After expanding Γ(τ) in a Taylor series around τ = 0, and equating 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, is found 2. Eqs. (7) and (8), with explicit expressions for S P, S S, S ρ, are plugged into the latter expression, and after some algebra kernels K P, K S and K ρ can be defined such that 3 δt = V [ ( ) ( ) ( )] δvp δvs δρ K P + K S K ρ d 3 x, (10) v P v S ρ and heteroengeities in Earth structure are now directly related to the delay time, i.e. something that we can pick from a seismogram. The functions K P (x), K S (x) and K ρ (x) have the peculiar banana-doughnut shape (with zero value on the infinite-frequency ray path) long discussed by Dahlen, Nolet and their co-workers at Princeton, and illustrated here in figures 1 and 2. Eq. (10) should be compared with eq. (2) of my lecture notes on body-wave tomography: they relate the same quantities, but the earlier equation rests on pure ray-theory approximation, and the integral there is to be performed only along the ray path; eq. (10) here should represent an improvement to that approximation. Some authors 4 have suggested that the application of finite-frequency tomography, as described here, to global body wave databases helps to enhance model resolution, imaging narrow 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 to define 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. 2 Dahlen, Hung and Nolet, GJI 2000, vol. 141 page 157, section 4.1. 3 The following equation is equivalent to eq. (77) of Dahlen and co-authors. 4 e.g., Montelli et al., Science, 303 pages 338ff, 2004.
Figure 1: Finite-frequency kernels for a uniform background medium. From Hung, Dahlen and Nolet, Fréchet kernels for finite-frequency traveltimes II. Examples, Geophysical Journal International, 141, pages 175ff, 2000.
Figure 2: Finite-frequency kernels for a spherically symmetric reference Earth model. From Hung et al., 2000.
Figure 3: P-velocity in the mantle from scattering theory: ascending plumes. From Montelli et 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) and the phase velocity of 150 s Love waves, for two different source-station geometries. From Boschi, Geophys. J. Int., 167, pages 238 252, 2006.
The problem can be collapsed to two dimensions only, like in the previous lecture (surface wave tomography). A (θ, φ)-dependent (not r-dependent) kernel relating δt(ω) and the corresponding phase velocity anomaly δc(θ, φ) is found at each frequency ω; let us call it K L (θ, φ; ω) for Love waves, or K R for Rayleigh waves; then eq. (9) from the previous lecture (surface wave tomography) is replaced by δt L (ω) = ω ray path K L(θ, φ; ω)δc L (θ, φ; ω)ds (11) (and an analogous expression for Rayleigh waves), and the inverse problem is set up exactly as in earlier lectures. Looking at K L (θ, φ; ω), K R (θ, φ; ω) we find them to be nonzero over very large portions of the Earth s surface, if compared to the body wave kernels, K P, etc. As surface waves are waves of lower frequency than body waves, it is to be expected that, in their case, limits of the high-frequency (ray theory) approximation be more evident, and scattering effects more relevant. Alternatively, one can leave explicit the dependence of surface wave phase on Earth structure at varying depth; an expression analogous to equation (17) of lecture 3 follows, with the kernels K SH, etc., now accounting for scattering effects as well. The former approach has been followed, for example, by R. Snieder and other authors from the Utrecht group 5, and likewise in the works of the ETH group illustrated in figures 5 and 6. In a more recent article 6, the Princeton group have worked out the latter, more general, approach. They show that known 2-D equations for phase velocity maps can be derived 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 normal modes. Hence, it does not involve ray paths. However, the requirement that heterogeneities be smooth is still needed. The rest of the Princeton treatment is substantially analogous to what I have outlined in this lecture, though much more detailed. 5 Snieder and Nolet, J. geophys. Res. 61 pages 55ff, 1987. Spetzler, Trampert and Snieder, Geophys. J. Int., 149 pages 755ff, 2002. 6 Zhou, Dahlen and Nolet, Geophys. J. Int., 158, pages 142ff, 2004.
Figure 6: The 2D inverse problem: phase-velocity maps obtained from inversions based on numerical finite-frequency (top) and ray theory (bottom), from Rayleigh-wave measurements at 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.