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The adjoint method in seismology – II. Applications:traveltimes and sensitivity functionals
A. Fichtner∗ , H.-P. Bunge, H. Igel,
Department of Earth and Environmental Sciences,Ludwig-Maximilians University Munich, Theresienstrasse 41, D-80333 Munich, Germany
Abstract
Sensitivity functionals which allow us to express the total derivative of a physical ob-servable with respect to the model parameters, are defined on the basis of the adjointmethod. The definition relies on the existence of Green’s functions for both the originaland the adjoint problem. Using the acoustic wave equation in a homogeneous and un-bounded medium, it is shown that the first derivative of the wavefieldu with respect tothe model parameterp = c−2 (c is the wave speed) does not contain traveltime informa-tion. This property translates to any objective function defined onu and in particular to theleast squares objective function. Therefore, a waveform inversion should either be com-plemented by a traveltime tomography or work with initially very long wavelengths thatdecrease in the course of the iteration. The definition of the sensitivity functionals naturallyintroduces waveform sensitivity kernels. Analytic examples are shown for the case of anisotropic, elastic and unbounded medium. In the case of a double couple source there arethree classes of sensitivity kernels, one for each of the parametersλ, µ (Lame parameters)andρ (density). They decompose into P→P, P→S, S→P and S→S kernels and can be de-scribed by third-order tensors incorporating the radiation patterns of the original wavefieldand the adjoint wavefield. An analysis of the sensitivity kernels for density suggests thata waveform inversion procedure should exclude either the direct waves or the source andreceiver regions. S-wave sensitivity kernels, corresponding to conversions from or to S-waves, are larger than the kernels corresponding to P-waves only. This implies that S-waveresiduals caused by parameter differences between the true Earth model and the numericalmodel will dominate a waveform inversion that does not account for that effect.
Key words: Inversion, Frechet derivative, waveform analysis, sensitivity
∗ Corresponding author.Email address:[email protected] (A. Fichtner).
Preprint submitted to Elsevier Science 23 February 2006
1 Introduction
Seismological observations play an important role in the imaging of the Earth’sinterior structure. Radial models of density and seismic velocities, based primarilyon the arrival times of seismic waves at the surface, were developed by Dziewon-ski & Anderson (1981), Kennett & Engdahl (1991) and Kennett et al. (1995). Theone-dimensional models reflect the static increase of pressure and temperature withincreasing depth and the associated phase transitions of mantle minerals. Moreover,they serve as reference models for maps of global three-dimensional variations ofdensity and seismic velocities obtained on the basis of ray theory and finite modesummations (e.g. Dziewonski, 1984; Grand, 1994; Masters et al., 1996; Grand etal., 1997; Kennett & Gorbatov, 2004). Today, it is possible to simulate the propaga-tion of seismic waves in realistic Earth models numerically (e.g. Igel et al., 1995;Komatitsch et al., 2000). This suggests that we replace ray theory and finite normalmode summations as forward models in inversion schemes by numeric solutions ofthe wave equation, thus, substantially increasing the amount of exploitable infor-mation.The inversion of seismic data can be based on different concepts. In probabilisticinverse theory the solution of an inverse problem is defined as a marginal probabil-ity distribution in the model parameter spaceP (Tarantola, 1987). This definitionis very general and elegant. However, a complete characterisation of this probabil-ity distribution usually requires us to evaluate a large number of parameter spaceelementsp ∈ P . Therefore, the probabilistic approach becomes impractical if themodel space is large and if the solution of the forward problem is time-consuming.As an alternative, the solution of an inverse problem may be defined as the parame-ter setpmin that minimises an objective functionE(u(p)) defined on the observableu(p). In symbols:pmin = {p ∈ P ; E(u(p)) = min}. Possible observables are ar-rival times or waveforms. Typical parameters are seismic velocities and density. Inthe context of the minimisation approach the sensitivity plays a central role. It isdefined as the first derivative of the observableu(p) with respect to its parametersp and is denoted byDpu. The sensitivity is the basis of resolution studies and min-imisation algorithms. Moreover, it influences the choice of data that one wishes toinclude in the inversion procedure.Often the computation ofDpu orDpE(u) by means of classical finite differencingtechniques is impractical due to the large number of parameters in realistic mod-els. A solution of this problem is the adjoint method (Tarantola, 1984, 1988). Itallows us to obtain the first derivative by solving the original forward problem andits adjoint problem only once, therefore being very efficient. As a consequence ofthe continuously increasing computation power the adjoint method has receivedmuch attention in many geo-scientific fields, such as for example meteorology (Ta-lagrand & Courtier, 1987), geodynamics (Bunge et. al, 2003), seismology (Craseet al., 1990; Igel et al., 1996; Tromp et al., 2005) or groundwater modelling (Sun,1994).For any application that employs the derivativeDpu it is important to know the type
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of information that it contains. This is particularly true for inversion algorithms thatexclusively rely onDpE(u) and therefore onDpu. Gauthier et al. (1986) used suchan algorithm (the method of steepest descent) for the purpose of acoustic wave-form inversion in two dimensions. They discovered that the derivative of the leastsquares objective function (the time integral over the squared waveform residuals)primarily contains information about where the edges of parameter perturbationsare located. However, the interior of their famous ’Camembert’ shaped bulk modu-lus perturbation could hardly be reconstructed during the inversion.A particularly interesting aspect of the adjoint method is that it allows us to findanalytic expressions for the different sensitivities (one for each parameter) in termsof Green’s functions. This directly leads to the formulation of sensitivity kernels,i.e., volumetric sensitivity densities. Asymptotic approximations for such sensitiv-ity kernels for body waves have been derived on the basis normal modes (Li &Tanimoto 1993; Li & Romanowicz 1995).The objective of this study is the analysis of sensitivity functionals, expressing thesensitivityDpu in terms of a functional that is independently linear in the sourcetime function and the model perturbations. Based on the example of the acousticwave equation in an unbounded and homogeneous medium, we show that the firstderivative of the least squares objective function only contains information aboutscatterers and reflectors. Traveltime information about a signal with a given wave-length is not contained in the first derivative. A simple numerical example in twodimensions confirms this result, which has important implications for any seismo-logical waveform inversion procedure that is founded on the gradient method alone.We find that even in the simple case of an unbounded, homogeneous, elastic medium,the waveform sensitivity kernels corresponding to density perturbations exhibit sur-prising complexity. Their shape depends on the source orientation, the observationdirection and the type of wave considered. In contrast to numerically computed ker-nels the analytic kernels automatically decompose into P→P, P→S, S→P and S→Skernels, meaning that they are sensitivity densities for different types of conversionsand therefore for different types of waveform residuals. The results derived froman analysis of the sensitivity kernels in such a simplified scenario yield importantinformation concerning the types of waves that one should consider for a waveforminversion.The long-term objective of this study is to contribute to the development of a wave-form inversion procedure based on the adjoint method and applicable on continentalor global scales. So far, waveform inversions applied to real data have mostly beenbased on simplified forward problems or relatively small parameter spaces (e.g.,Cichowicz & Green 1992; Marquering et al. 1996; Zielhuis & van der Hilst 1996).
Before we proceed, we will briefly review the principal results obtained in the firstof this series of two papers on the adjoint method in seismology (Fichtner et al.,2005). Letu be a field defined in time and space and depending on a set of param-etersp, i.e., u = u(p;x, t). (In what follows, vectorial and potentially vectorialquantities are in bold face.) We assume thatu is determined through a differentialequation complemented by an adequate set of subsidiary conditions. Alternatively
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and equivalently one may consider adifferential operatorL mappingu onto theright-hand sideh of the differential equation:
L(u;p,x, t) = h(x, t) . (1)
The operatorL itself may depend on timet, spacex and the parameter setp.Solving the differential equation means to find an inverse ofL. It is usually anobjective functionE(u,p) defined onu andp, rather thanu itself that is of interest.In seismology,E may for example be a time integral over the difference betweenan observed wavefieldu0 and a synthetically generated wavefieldu. Here, we shallassume thatE can be written as
E(u,p) =∫
G
∫ t1
t=t0f(u,p) d3x dt =
∫G
∫ t1
t=t01 · f(u,p) d3x dt =: 〈1, f(u,p)〉 ,
(2)whereG is the region of interest (e.g. the Earth) and[t0, t1] the temporal observationinterval. The functionf must be specified for particular applications. Note thatfor reasons of greater generality the objective functionE is assumed to dependexplicitly on both the observable and the parameters. Restricting our attention tolinear operatorsL, we found that the total derivative ofE with respect to the modelparametersp in the directionp′ can be expressed as
DpE(u,p)(p′) = 〈p′, ∂pf(u,p) + Lp(u,ψ)〉 , (3)
whereLp is theparameter transposeof L. Theadjoint fieldsψ has to satisfy theadjoint equation
Lu(ψ;p,x, t) + ∂uf = 0 , (4)
and a set ofadjoint subsidiary conditions. The symbolLu denotes the transpose ofL with respect to the observable. For the two-dimensional scalar wave equation theoperatorL is
L(u; ρ, µ,x, t) = ρ∂2t u−∇ · (µ∇u) , (5)
The symbolsρ andµ represent the density and shear modulus distributions, respec-tively. The corresponding adjoint equation coincides with the original wave equa-tion (5) because there is no dissipation involved. Moreover, the spatial boundaryconditions translate one-to-one from the original problem to the adjoint problem,whereas the initial conditions translate to terminal conditions. Using the adjointwavefieldψ one can compute the derivative of the objective functionE = 〈1, f〉 :
DpE(u)(p′) = 〈ρ′, ∂ρf−∂tψ ∂tu〉+〈µ′, ∂µf+(∇u)·(∇ψ)〉 , p′ = (ρ′, µ′) . (6)
In the case of the elastic wave equation with attenuation the operatorL is
L(u;Φ, ρ,x, t) = ρ(x)∂2t u(x, t)−∇ ·
∫ ∞
−∞Φ(x, t− τ) : ∇u(x, τ) dτ , (7)
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with Φ representing the fourth-order rate of relaxation tensor. The adjoint equationfor this problem is found to be
Lu(ψ;Φ, ρ,x, t) = ρ(x)∂2tψ(x, t)−∇ ·
∫ ∞
−∞Φ(x, τ − t) : ∇ψ(x, τ) dτ .
(The sets of subsidiary conditions for both the original and the adjoint problem areomitted here for brevity.) The operatorL is not symmetric, meaning that it is notidentical with its transposeLu. Again, using the adjoint wavefieldψ one can obtainthe derivative ofE = 〈1, f〉 with respect to the parametersp :
D(Φ,ρ)E(u,Φ, ρ)((Φ′, ρ′)) = DρE(u,Φ, ρ)(ρ′) +DΦE(u,Φ, ρ)(Φ′)
=∫
G
∫ t1
t=t0ρ′(x)
[∂ρf +ψ(x, t) · ∂2
t u(x, t)]dt d3x
+∫
G
∫ t1−t0
t=0Φ′(x, t) ::
[∂Φf +
∫ t1
τ=t0+t∇ψ(x, τ)⊗∇u(x, τ − t) dτ
]dt d3x .
(8)
The symbol⊗ denotes the tensor or dyadic product ((a ⊗ b)ijkl = aijbkl) and thesymbol:: the quadruple scalar product (A :: B = AijklBijkl).
2 Definition of the sensitivity functional
We define the sensitivitys to be the derivative of the physical observableu =u(p;x, t) with respect to the model parametersp in the directionp′ at some timet = τ0 and locationx = ξ0. In symbols:
s(ξ0, τ0,p′) := Dpu(p; ξ0, τ0)(p
′) . (9)
The sensitivity is of particular importance when the solution of an inverse problemin the minimisation sense can be approximated with gradient methods. They relyon the derivative of the objective functionE with respect to the model parametersp, which is given by
DpE(u,p)(p′) = ∂uE(u,p)(Dpu)+∂pE(u,p)(p′) = ∂uE(u,p)(s)+∂pE(u,p)(p′) .(10)
The sensitivitys is the only contribution toDpE that is directly due to the changeof the observableu arising from a perturbation of the model parametersp.In order to obtain an expression fors, we choose a specific objective functionEs
which is equal to thei−component of the wavefield observed atx = ξ0 andt = τ0:
Es(u,p) = Es(u) = ei · u(p; ξ0, τ0) = ui(p; ξ0, τ0) , i = 1, 2 or 3. (11)
Then the derivative ofEs equals the derivative of thei−th wavefield component,that isDpEs(u)(p′) = Dpui(p
′) = si. When the parameter spaceP has a finite
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dimensionm <∞ and basis vectorsek, with k = 1, ...,m, then we find
DpEs(u)(ek) = Dpui(p; ξ0, τ0)(ek) = 〈ek,Lp(u,ψ)〉 =
d
dpk
ui(p; ξ0, τ0) , (12)
wherepk is thek−component of the elementp ∈ P . Thus, the total derivative ofEs coincides with the(i, k)− component of the parameterJacobian. Our specificchoice of the objective functionEs implies that the generatorfs that linksEs to thebilinear form〈 . , . 〉 (see equation (2)) is determined through
fs(u,p) = fs(u) = ei · u(p;x, t) δ(x− ξ0) δ(t− τ0) , (13)
Therefore the adjoint equation is
Lu(ψ;p,x, t) = −∂ufs(x, t) = −ei δ(x− ξ0) δ(t− τ0) . (14)
The adjoint source is restricted to a single point in time and space, namely to theobservation timeτ0 and the observation pointξ0. Hence, by definition,ψ is thei−th Green’s function of the adjoint problem multiplied by−1,
ψ(p;x, t) = −g∗i (p; ξ0, τ0;x, t) , (15)
and thereforesi(ξ0, τ0;h,p
′) = −〈p′,Lp(u,g∗i )〉 . (16)
Consequently, one can compute Jacobians if the Green’s functions of the adjointproblem are known. In the case of linear equations, the outstanding property ofGreen’s functions is that they allow us to represent any solution by means of asimple convolution with the source functionh (equation (1)). Here we leth be aweighted superposition of spatial point sources with different directions and causalsource time functionh:
h(x, t) =3∑
k=1
αkek δ(x− x0)h(t− t0) , h(t)|t<0 = 0 , αk ∈ R . (17)
Thenu is determined through
u(x, t) =3∑
k=1
αk
∫ ∞
−∞gk(p;x0, t0;x, t− τ)h(τ) dτ . (18)
Combining equations (16) and (18) yields
si(ξ0, τ0;h,p′)
= −3∑
k=1
αk
∫ ∞
−∞〈p′,Lp[gk(p;x0, t0;x, t− τ),g∗i (p; ξ0, τ0;x, t)]〉h(τ) dτ . (19)
The sensitivity componentsi can be represented by a functionalSki which is inde-pendently linear with respect top′ andh, respectively. We will call this operator
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thesensitivity functional:
Ski(ξ0, τ0;h,p′) := −
∫ ∞
−∞〈p′,Lp[gk(p;x0, t0;x, t−τ),g∗i (p; ξ0, τ0;x, t)]〉h(τ) dτ .
(20)The symbol:= denotes equality by definition. Thus,
si(ξ0, τ0;h,p′) = Dpui(p; ξ0, τ0)(p
′) =3∑
k=1
αkSki(ξ0, τ0;h,p′) . (21)
Here we considered only one receiver and one vectorial source. It is howeverstraightforward to extend the concept of sensitivity functionals to multiple source/receivergeometries and to fields set off by a dipolar source.
3 Example: The acoustic wave equation
This section concentrates on the sensitivity functional for an acoustic wavefield ina homogeneous medium. Even though the problem is oversimplified, many of theresults translate to more complex cases that include elasticity and heterogeneity.
3.1 Sensitivity functional
The acoustic wave equation for the case of small density variations is given by
L[u; p] = p(x) ∂2t u(x, t)−∆u(x, t) = f(x, t) , (22a)
with x ∈ R3 and t ∈ [t0, t1] ⊂ R. The parameterp is the squared inverse ofthe phase velocity:p = c−2. We complement equation (22a) by the regularity andinitial conditions
lim|x|→∞
u(x, t) = lim|x|→∞
∂tu(x, t) = 0 , (22b)
u(x, t)|t≤t0 = ∂tu(x, t)|t≤t0 = 0 . (22c)
The Green’s function for equations (22) in the case ofp = const is
g(p;x0, t0;x, t) =1
4π
δ((t− t0)− |x− x0|/c)|x− x0|
. (23)
Since the acoustic wave equation operator is symmetric (e.g. Tarantola 1984), theadjoint equation forg∗ is
p(x)∂2t g
∗(p; ξ0, τ0;x, t)−∆g∗(p; ξ0, τ0;x, t) = δ(x− ξ0)δ(t− τ0) (24a)
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with the subsidiary conditions
lim|x|→∞
g∗(p; ξ0, τ0;x, t) = lim|x|→∞
∂tg∗(p; ξ0, τ0;x, t) = 0 , (24b)
g∗(p; ξ0, τ0;x, t)|t≥t1 = ∂tg∗(p; ξ0, τ0;x, t)|t≥t1 = 0 . (24c)
In equations (24),ξ0 is the observation point andτ0 ∈ [t0, t1] is the observationtime of the original acoustic fieldu. The solution of equations (24) forp = constis
g∗(p; ξ0, τ0;x, t) =1
4π
δ((τ0 − t)− |x− ξ0|/c)|x− ξ0|
. (25)
It remains to assemble the building blocks (23) and (25) to form the sensitivityfunctionalS. The expression for the parameter transposeLp is the same as in thetwo-dimensional case. Therefore, we find
S(ξ0, τ0;h, p′) = −
∫ ∞
−∞〈p′, Lp[g(p;x0, t0;x, t− τ), g∗(p; ξ0, τ0;x, t)]〉h(τ) dτ
= −∫ ∞
τ=−∞
∫ t1
t=t0
∫R3p′(x)g∗(p; ξ0, τ0;x, t)∂
2t g(p;x0, t0;x, t− τ)h(τ) dτ dt d3x
= − 1
(4π)2
∫ t1
t=t0
∫R3p′(x)δ[(τ0 − t)− |x− ξ0|/c]
h(t− t0 − |x− x0|/c)|x− ξ0||x− x0|
d3x dt.
(26)
The integral overt is non-zero only if the condition
τ0 ∈ [t0 + |x− ξ0|/c , t1 + |x− ξ0|/c] (27)
is satisfied. (Note that we already requireτ0 ∈ [t0, t1].) If relation (27) indeed holds,we obtain
S(ξ0, τ0;h, p′) = − 1
(4π)2
∫R3
p′(x)h[τ0 − t0 − (|x− x0|+ |x− ξ0|)/c]
|x− ξ0||x− x0|d3x .
(28)This is the derivative of the displacement field of an acoustic wave in a homoge-neous medium with respect top = c−2.
3.2 Traveltime information
Due to the causality ofh the sensitivity functional is non-zero only if the relation
c(τ0 − t0) ≥ |x− x0|+ |x− ξ0| (29)
can be satisfied for somex ∈ R. This is possible only forc(τ0 − t0) ≥ |x0 − ξ0|,becausec(τ0 − t0) < |x0 − ξ0| would violate the triangle inequality. Thus,
Dpu(p; ξ0, τ0)(p′) = 0 , ∀ τ0 < t0 + |x0 − ξ0|/c . (30)
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In other words: The derivative of the acoustic wavefieldu with respect to the pa-rameterp = c−2 in any directionp′ at some observation pointξ0 is identically zeroprior to the first arrival which occurs at the timet = t0 + |x0−ξ0|/c. Consequently,we can perturb the medium arbitrarily andDpu(p; ξ0, τ0)(p
′) will always be zeroprior to the first arrival atx = ξ0. However, we know that increasing the wavespeedsomewhere between sourcex0 and receiverξ0 will lead to earlier arrivals.The resolution of this apparent paradox imposes significant constraints on the ap-plicability of pure gradient methods in the field of waveform inversion. Assumethat the wave fieldu(p;x, t) is sufficiently smooth with respect top to allow itsexpansion into a Taylor series,
u(p+ p′;x, t) = u(p;x, t) +Dpu(p;x, t)(p′) +
1
2D2
pu(p;x, t)(p′, p′) + ... (31)
(Note that the perturbationp′ = p′(x) does not need to be restricted to a point. Itcan be spatially extended.) The unperturbed fieldu(p;x, t) and the perturbed fieldu(p+ p′;x, t) are both governed by an acoustic wave equation
p ∂2t u(p;x, t)−∆u(p;x, t) = δ(x− x0)h(t− t0) , (32)
(p+ p′) ∂2t u(p+ p′;x, t)−∆u(p+ p′;x, t) = δ(x− x0)h(t− t0) . (33)
Combining equations (32) and (33) and inserting the Taylor expansion ofu, weobtain, correct to first order in||p′||,
p ∂2tDpu(p;x, t)(p
′)−∆Dpu(p;x, t)(p′) = −p′∂2
t u(p;x, t) . (34)
Equation (34) states that correct to first order in||p′|| the perturbed wave fieldu(p+p′;x, t) is given by the sum of the unperturbed wave fieldu(p;x, t) and the wavefield Dpu(p;x, t)(p
′). The latter is set off by secondary sources that act when theunperturbed wave field reaches the perturbationp′(x). The time when the sourcesfor Dpu(p;x, t)(p
′) act, is therefore independent of the actual perturbationp′(x).Consequently, correct to first order, the wave fieldDpu(p;x, t)(p
′) must arrive laterat some pointx = ξ0 than the unperturbed fieldu(p;x, t). The first derivative ofthe displacement fieldu with respect to the model parameters only accounts forscattering caused by finite perturbations. This implies that traveltime differencesmust be contained in higher derivatives. Hence, one may symbolically write
u(p+ p′;x, t)− u(p;x, t) = scattering (1st derivative)+ travel time differences (higher derivatives). (35)
Note that equation (10) states that this property also influences the derivative of anyobjective functionE(u,p) with respect to the parameters. This phenomenon alsomanifests itself in a simple numerical example (see figure 1) that is based on a finitedifferences solution of the two-dimensional scalar wave equation
ρ(x) ∂2t u(x, t)−∇ · (µ(x)∇u(x, t)) = g(x, t) , x ∈ G ⊂ R2 (36)
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Fig. 1.Left : Perturbed density model. The reference model is homogeneous in both densityand shear modulus (ρ = 3.0 · 103 kg m−3, µ = 75.0 · 109 N m−2). There are no shear mod-ulus perturbations.Middle : Derivative of the objective functionE with respect to densityfor a dominant signal wavelength ofλ0 = 14 km. Only the edges of the perturbation aremapped byDρE. The interior of the density perturbation remains invisible. (The colorbaronly applies to the figure on the left because the amplitude of the derivatives are propor-tional to the amplitudes of the signal.)Right: DρE for λ0 = 40 km. Since the wavelengthis longer, the entire density perturbation acts as a scatterer. It therefore maps as one singleobject into the derivative.
in a square region. The parametersρ andµ are density and shear modulus, respec-tively. The upper boundary is free (no tractions), the lower boundary is rigid (nodisplacement) and periodic boundary conditions are implemented on the left andright boundaries. Ten receivers at a mutual distance of10 km are located alongthe upper boundary. Their positions coincide with those of ten sources which radi-ate a Ricker wavelet. Homogeneous density and shear modulus distributions (ρ =3.0 · 103 kg m−3, µ = 75.0 · 109 N m−2) serve as references. The perturbed modelcontains in its centre a square box where the density is reduced to a value of2.75 · 103 kg m−3 (see figure 1, left). The shear modulus distributionµ(x) remainsunchanged. A time integral over the sum of all squared residuals is used as objectivefunctionE:
E(u) =1
2
10∑k=1
∫ t1
t=t0[u(xk, t)− u0(xk, t)]
2 dt . (37)
The wavefieldsu andu0 correspond to the perturbed and the unperturbed mod-els, respectively. As expected, the derivative ofE with respect to density dependsstrongly on the dominant wavelengthλ0 of the signal. Choosingλ0 = 14 km resultsin the derivative shown in the middle of figure 1. Rather than mapping the entirebox, the derivative is large only along its upper and lower edges. Those parts of theperturbation act as reflectors for signals with a wavelength that is short comparedto the extension of the perturbation. Since it depends on the wavelength whethera perturbation acts completely as scatterer or not, it is possible to map the entiredensity perturbation by choosing a longer wavelength. The derivative ofE with re-spect to density for the wavelengthλ0 = 40 km is shown on the right of figure 1.As predicted, the density perturbation is mapped in the form of a single gradientcontribution because it acts as one scatterer for this particular wave. Naturally, thegradient peak is much broader due to the longer wavelength.
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A profound implication arising from the above considerations is that a waveforminversion based exclusively on the first derivative of the wavefield or any associ-ated objective function is not well suited for the determination of structure thathas a significantly longer wavelength than the signal. Consequently, the waveforminversion must either be based on a more sophisticated iterative procedure that in-cludes higher derivatives or it must be complemented by a conventional traveltimetomography. That a waveform inversion requires a background model that explainstraveltimes sufficiently well has also been found by Crase et al. (1990) and Igel etal. (1996).
4 Waveform sensitivity kernels in homogeneous isotropic media
Based on equation (20) one may approximate sensitivity functionals numericallyfor almost arbitrarily complex media. However, in order to gain physical under-standing it is important to analyse simplified problems that can be solved analyti-cally. In the particular case of the wave equation in a homogeneous, isotropic andunbounded medium, it is possible to demonstrate the complexities arising fromwave conversions which are due to perturbations in different parameters.Consider the operatorL corresponding to the general wave equation (equation (7)).By inserting the time-localised and isotropic constitutive relation
Φ(x, t) = Φ0(x)δ(t) , Φ0ijkl = λδijδkl + µδikδjl + µδilδjk , (38)
the operatorL and its displacement transposeLu reduce to
L[u; ρ, λ, µ,x, t] = ρ∂2t u− (2µ+ λ)∇∇ · u + µ∇× (∇× u) (39)
and
Lu[ψ; ρ, λ, µ,x, t] = ρ∂2tψ − (2µ+ λ)∇∇ ·ψ + µ∇× (∇×ψ) . (40)
The Lame parametersλ andµ are assumed constant. Since the time-localised rhe-ology eliminates anelasticity,L becomes symmetric. Also based on (38), one findsthat the total derivative of the least squares objective functionE with respect to theparametersp = (ρ, λ, µ) and in the directionp′ = (ρ′, λ′, µ′) simplifies to
DpE(u)(p′) =−∫
R3
∫ t1
t=t0ρ′ ∂tψ · ∂tu dt d
3x +∫
R3
∫ t1
t=t0λ′(∇ ·ψ)(∇ · u) dt d3x
+∫
R3
∫ t1
t=t0µ′[(∇ψ) : (∇u) + (∇ψ) : (∇u)T ] dt d3x . (41)
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Here it is assumed thatE does not explicitly depend onp. The sources of mosttectonic earthquakes can be approximated by a superposition of dipoles
fpq(x, t) = (eq · ∇x0)δ(x− x0)h(t− t0) ep
= limε→0
1
ε[δ(x− x0 + εeq) ep − δ(x− x0) ep]h(t− t0) , (42)
whereh is a causal source time function. Withgp(x0, t0;x, t) the Green’s functionfor a single spatio-temporal point force acting atx = x0 andt = t0 in the directionof ep, the solution for the(p, q)−dipolefpq is
gpq(x0, t0;x, t) := (eq · ∇x0)gp(x0, t0;x, t) . (43)
The symbol:= denotes equality by definition. A general displacement field result-ing from a weighted superposition of all possible dipoles can then be expressed inthe form
u(x, t) =3∑
p,q=1
Mpq
∫ ∞
−∞gpq(x0, t0;x, t− τ)h(τ) dτ , (44)
where the scalarsMpq are the moment tensor components. A comparison with equa-tions (18) to (21) suggests that the appropriate definition of the sensitivity func-tional for the case of a dipolar source is given by the following set of equations:
sn(ξ0, τ0;h,p′) =
3∑p,q=1
MpqSpqn(ξ0, τ0;h,p′) , (45a)
Spqn(ξ0, τ0;h,p′) = Sρ
pqn(ξ0, τ0;h, ρ′) + Sλ
pqn(ξ0, τ0;h, λ′) + Sµ
pqn(ξ0, τ0;h, µ′) ,
(45b)
Sρpqn(ξ0, τ0;h, ρ
′) =
∞∫−∞
t1∫t=t0
∫R3
ρ′∂tgpq(x0, t0;x, t− τ)
· ∂tg∗n(ξ0, τ0;x, t)h(τ) d
3x dt dτ , (45c)
Sλpqn(ξ0, τ0;h, λ
′) = −∞∫
−∞
t1∫t=t0
∫R3
λ′ [∇ · gpq(x0, t0;x, t− τ)]
· [∇ · g∗n(ξ0, τ0;x, t)] h(τ) d3x dt dτ , (45d)
Sµpqn(ξ0, τ0;h, µ
′) = −∞∫
−∞
t1∫t=t0
∫R3
µ′[∇gpq(x0, t0;x, t− τ)]
:[∇g∗n(ξ0, τ0;x, t)]h(τ) d3x dt dτ ,
−∞∫
−∞
t1∫t=t0
∫R3
µ′[∇gpq(x0, t0;x, t− τ)]
:[∇g∗n(ξ0, τ0;x, t)]T h(τ) d3x dt dτ . (45e)
12
Due to the simplicity of the problem it is possible to derive exact expressions forthe dipole Green’s functions (e.g. Aki & Richards, 2002). Restricting our consid-erations to the far field we may writegpq as a sum of an S-wave and a P-wavecontribution:
gPFFpq (x0, t0;x, t) =
γ(s)p γ(s)
q
4πρα3r(s)
r(s)
r(s)δ
(t− t0 −
r(s)
α
), (46a)
gSFFpq (x0, t0;x, t) =
γ(s)q
4πρβ3r(s)
(ep −
r(s)
r(s)γ(s)
p
)δ
(t− t0 −
r(s)
β
). (46b)
TheP−wave speedα and theS−wave speedβ are√
(2µ+ λ)/ρ and√µ/ρ, re-
spectively. The distance from the source isr(s) = |x − x0| andγ(s)i = r
(s)i /r(s) is
the direction cosine measured from the source. The superscriptXFF with X = Por X = S is a reminder that this is a far field approximation. As a final buildingblock we require the adjoint Green’s functions. Since the adjoint wave equationcoincides with the original wave equation in the case of perfect elasticity, the ad-joint Green’s functionsg∗n are identical to the original Green’s functions, the onlydifference being that theg∗n have to satisfy the terminal conditions. Therefore wefind
g∗PFFn (ξ0, τ0;x, t) =
1
4πρα2r(r)
r(r)
r(r)γ(r)
n δ
(r(r)
α+ t− τ0
), (47a)
g∗SFFn (ξ0, τ0;x, t) =
1
4πρβ2r(r)
(en −
r(r)
r(r)γ(r)
n
)δ
(r(r)
β+ t− τ0
). (47b)
with r(r) = (x − ξ0) andγ(r)i the corresponding direction cosines measured from
the receiver. Again, this is a far field approximation, which is justified when the pa-rameter perturbationsρ′, λ′ andµ′ are several wavelengths away from both sourceand receiver. However, if a perturbation is close to the receiver it becomes neces-sary to include the near field terms in the adjoint Green’s functionsg∗n. Similarly,perturbations near the source will require the near and intermediate field terms ingpq. Consequently, both the source and the receiver regions have to be excludedfrom the considerations in the case of the far field approximation.In the appendix it is demonstrated that the sensitivity functional for density pertur-bations can be reduced to
Sρpqn(ξ0, τ0;h, ρ
′) =−∫
R3aPFF
pq · bPFFn
...h(τ0 − t0 − r(s)/α− r(r)/α
)d3x
−∫
R3aPFF
pq · bSFFn
...h(τ0 − t0 − r(s)/α− r(r)/β
)d3x
−∫
R3aSFF
pq · bPFFn
...h(τ0 − t0 − r(s)/β − r(r)/α
)d3x
−∫
R3aSFF
pq · bSFFn
...h(τ0 − t0 − r(s)/β − r(r)/β
)d3x .
(48)
13
The contributions for the volumetric sensitivity kernelsaXFFpq andbY FF
n , whereXandY can beP or S, are given by
aPFFpq =
γ(s)p γ(s)
q
4πρα3r(s)
r(s)
r(s), aSFF
pq =γ(s)
q
4πρβ3r(s)
(ep −
r(s)
r(s)γ(s)
p
),
bPFFn =
γ(r)n
4πρα2r(r)
r(r)
r(r), bSFF
n =1
4πρβ2r(r)
(en −
r(r)
r(r)γ(r)
n
). (49)
Evidently, these factors just coincide with the radiation patterns of the original andthe adjoint Green’s functions, respectively. In the case of perturbations in Lame’sparameters the corresponding expressions will become more complex. Note thatthe causality of the source time functionh implies that all sensitivity functionalsfor density with kernels
RX→Ypqn := −aXFF
pq · bY FFn (50)
are identically zero if the observation timeτ0 does not satisfy the condition
τ0 ≥ t0 + r(s)/cX + r(r)/cY , cp = α , cs = β . (51)
This is again a manifestation of the fact that the first derivative ofE and thereforethe sensitivity functionals do not contain traveltime information. The observationtime has to be larger than the time required for a wave in the unperturbed medium totravel from the sourcex0 to a perturbation atx and from there to the receiver atξ0.The volumetric sensitivity kernels themselves are not dependent on time. However,only those regions of them are sampled where the expression
...h(τ0 − t0 − r(s)/β − r(r)/β) (52)
is non-zero. The existence of four kernels is the consequence of possible conver-sions from P-waves to S-waves and vice versa.
Superpositions of two dipoles (double couples) are of particular interest in seismol-ogy. They can be used to approximate sources corresponding to slip across a faultplane. The number of possible sensitivity kernels in the case of a double couplesource is36 for each one of the parametersρ, µ andλ. There are3 observationdirectionsn (n = 1, 2, 3), 3 distinct double couples ((p, q) = (1, 2), (1, 3), (2, 3)),and4 different types of kernels (P→P, P→S, S→P, S→S). Figures (2) to (5) showiso-surfaces of the36 kernels corresponding to density perturbations. They areplotted for values of±s · [(600 km)2 · (4πρc3pq) · (4πρc2n)]−1, where the scalingfactor s is given in the sub-figure titles. The velocitycpq is α = 5 km s−1 for theP→Y kernels andβ = 5/
√2 km s−1 for the S→Y kernels. Similarly,cn is α for
the X→P kernels andβ for the X→S kernels. Source and receiver are located at(x, y, z) = (200, 500, 500) km and(x, y, z, ) = (800, 500, 500) km, respectively.Black lines intersecting at the source indicate the directionsp andq.Despite the simplicity of the problem, the sensitivity densities reveal a considerable
14
Fig. 2. Iso-surfaces of the sensitivity kernels−aPFFpq · bPFF
n plotted at values of±s · [(600 km)2 · (4πρα3) · (4πρα2)]−1, where the scaling factors is given in the sub-figuretitles. Yellow denotes positive and blue negative values. Wave speeds areα = 5 km s−1
andβ = α/√
2 km s−1. The source is located at(x, y, z) = (200, 500, 500) km and thereceiver, marked by a star (*), is at(x, y, z) = (800, 500, 500) km. Black lines intersectingat the source indicate the(pq)-plane. The indexn represents the observation direction. Thesensitivity kernels plotted in the first row ((1, 2)-couple) have expected correspondences inthe second row ((1, 3)-couple). It is noteworthy that there are significant sensitivity contri-butions between source and receiver only if one of the two source dipoles points towardsthe receiver. Therefore, in the case of the(2, 3)-couple the sensitivities practically vanishin the region between source and receiver. Large sensitivity contributions in regions behindsource or receiver and far away from the line connecting source and receiver arise if theobservation directionn does not point toward the source.
complexity. As expected, they depend on both the observation directionn and thesource orientation(pq). In general, all the sensitivity densities are small along theline connecting source and receiver. This has important consequences: Assume thatwe observe a waveform difference in the direct wave (either P or S) that is due toa density perturbation. (Of course, in practice we may not know the character ofthe perturbation. This is a gedanken experiment.) Then we know that this density
15
Fig. 3. The same as figure 2 but for the sensitivity kernels−aPFFpq · bSFF
n (P to S conver-sions) plotted at values of±s · [(600 km)2 · (4πρα3) · (4πρβ2)]−1. The scaling factorsis given in the sub-figure titles. Symmetries between the first row ((1, 2)-couple) and thesecond row ((1, 3)-couple) are again due to choosing the line connecting source and re-ceiver as parallel to thex-direction. In general the sensitivity densities are small near theregion between source and receiver. This effect is especially pronounced in the case of the(2, 3)-couple, as already observed for the P→P kernels shown in figure 2.
perturbation must be located somewhere along the line (or more generally the ray)connecting source and receiver. If it were not located on that line, the secondarywave emanating from that density perturbation would arrive later and therefore nottranslate to a waveform change of the direct wave, at least if its temporal bandwidthis sufficiently short. However, the sensitivity kernels for density perturbations aresmall or even zero on that line. But there are two exceptions, namely the source andthe receiver regions where all sensitivity kernels have a singularity. Thus, we canconclude that the density perturbation is very likely to be located near the source orreceiver because if it were located somewhere else along the source-receiver line, itwould have to have a very large and probably unrealistic amplitude. Consequently,one may want to exclude direct waves from a waveform inversion because the cor-responding residuals will map almost exclusively into the source and receiver re-gions. Alternatively, one could use the direct wave but at the same time exclude thesource and receiver regions from the inversion by setting the derivatives there tozero.Another important aspect of figures 2 to 5 is their scaling. The values of the P→Pkernels are proportional toα−3α−2, whereas the other kernels have values propor-
16
Fig. 4. The same as figure 2 but for the sensitivity kernels−aSFFpq · bPFF
n (S to P con-versions) plotted at values of±s · [(600 km)2 · (4πρβ3) · (4πρα2)]−1. The scaling factors is given in the sub-figure titles. As expected from the source-receiver asymmetry of thesensitivity kernels, there are no similarities between the S→P kernels and the P→S kernelsshown in figure 3. Generally, the sensitivity densities are small in the region between sourceand receiver, the only exceptions being the(121) and the(131) kernel.
tional to α−3β−2 (P→S), β−3α−2 (S→P) andβ−3β−2 (S→S), respectively. Theimplication for any waveform inversion is that S-wave residuals dominate and thatP-wave residuals are suppressed, therefore potentially leading to an unbalanced ex-ploitation of information contained in a seismogram. Waveform residuals due to aperturbation inλ will be comparatively small because they are pure P-wave resid-uals.In figure 2 we observe that the(1, 2)-double couples correspond to kernels for the(1, 3)-double couples with different observation directions. This symmetry arisesfrom choosing the source-receiver line parallel to thex-direction. Similar corre-spondences can be found for the P→S, S→P and S→S kernels. Since the sensitivitykernels for density are just a product of the radiation pattern of the source doublecouple and the radiation pattern of a single force pointing in the observation direc-tion, they can directly be interpreted in terms of these patterns. The(121)-kernelsin figures 2 and 3 must vanish on the source-receiver line because the P-wave am-plitude along this line is zero.Similar arguments can be found for the S→P kernels in figure 4. The(1, 2)- and(1, 3)-double couples do generate S-wave motion between source and receiver.However, a P-wave set off by an S-wave incident on a density perturbation lo-
17
Fig. 5. The same as figure 2 but for the sensitivity kernels−aSFFpq · bSFF
n plotted at valuesof ±s · [(600 km)2 · (4πρβ3) · (4πρβ2)]−1. The scaling factors is given in the sub-figuretitles. Again, most of the sensitivity densities are concentrated in regions around source orreceiver. Significant contributions along the line connecting source and receiver can onlybe observed in the cases of the(122) and(133) kernels.
cated somewhere on the source-receiver line, has zero amplitude along that line.This is because the incident S-wave acts as a single force in they- or z-direction,respectively.Generally, the sensitivity kernels corresponding toλ andµ do not allow such sim-ple interpretations because they are the products of more complicated patterns.At this point one may conclude that one general feature of the density sensitiv-ity kernels involving S-waves (primary or secondary) is that they are collectivelysmall or even zero between source and receiver. So far however, we consideredonly very specific geometries, namely those where the dipole directions coincidewith the coordinate directions. A different set up is presented in figure 6 for thecase of P→S kernels. The source-receiver line has been rotated by45o, so thatthe source is now located at (x, y, z) = (300, 300, 500) km and the receiver is at(x, y, z) = (724, 724, 500) km. Despite this modification, the P→S kernels shownin figure 6 do not differ much from those presented in figure 3. Only the size of thelobes changes and they are curved towards either source or receiver. The generalfeature, namely the fact that they vanish in the region between source and receiver,remains.Finally, we consider the sensitivity kernels corresponding to the parametersµ andλ. A perturbation inµ is expected to set off both a secondary S- and a secondary
18
Fig. 6. The same as figure 3 but with the source and the receiver located at(x, y, z) = (300, 300, 500) km and(x, y, z) = (724, 724, 500) km, respectively. Symme-tries between the first row ((1, 2)-couple) and the second row ((1, 3)-couple) as observedin figure 3 do not exist. In general the sensitivity densities are small in the region be-tween source and receiver. This effect is again especially pronounced in the case of the(2, 3)-couple.
P-wave. In particular we find that the far field sensitivity kernels forµ are given by
MX→Ypqn = −AXFF
pq : BY FFn −AXFF
pq :(BY FF
n
)T, (53)
whereX andY denote eitherS or P . It is demonstrated in the appendix that thecartesian components of the second-order tensorsAXFF
pq andBY FFn are
(APFF
pq
)ij
=γ(s)
p γ(s)q γ
(s)i γ
(s)j
4πρα4r(s)
(ASFF
pq
)ij
=γ(s)
q γ(s)i
4πρβ4r(s)
(δjp − γ
(s)j γ(s)
p
),
(BPFF
n
)ij
=γ
(r)j γ
(r)i γ(r)
n
4πρα3r(r),
(BSFF
n
)ij
=γ
(r)i
4πρβ3r(r)
(δjn − γ
(r)j γ(r)
n
).
(54)
It is not possible to interpret these kernels simply in terms of multiplied radiationpatterns. Their maximum values are a factor ofα−1β−1 smaller than the maximumvalues of the sensitivity kernels for density.
From equation (45d) we see that a perturbation inλ will not create any secondaryfar field S-waves because the divergence of the adjoint S-wave Green’s function
19
has no far field contribution. Since the same is true for the original S-wave Green’sfunction, there is only one sensitivity kernel forλ for the different double coupleorientations and observation directions:
LFFpqn := −aFF
pq bFFn . (55)
The scalarsaFFpq andbFF
n are given by
aFFpq =
γ(s)p γ(s)
q
4πρα4r(s), bFF
n =γ(r)
n
4πρα3r(r). (56)
See the appendix for a detailed derivation. The general characteristics of the sensi-tivity kernels for density translate to the kernels forµ andλ. In particular, they aresmall in the region between source and receiver.
5 Discussion
The sensitivitys is defined as the first derivative of an observableu(p) with respectto the model parametersp. It can be expressed by the Green’s functions of the orig-inal and the adjoint problem (e.g. Tarantola, 1984, 1988; Talagrand & Courtier,1987; Sun, 1994; Bunge et al., 2003). An analysis for the case of the acoustic waveequationp ∂2
t u−∆u = f led to the result that the sensitivitys(ξ0, τ0) vanishes fortimesτ0 that are smaller than the arrival time of the first wavefront at the observa-tion pointξ0. This is true for all possible differentiation directions (perturbations)δp(x) and in particular for those that would lead to earlier arrivals. Informationabout traveltime anomalies must therefore be contained in higher derivatives of thewavefield.This characteristic influences the total derivative of an objective functionE with re-spect top through equation (12). WhenE is chosen to be the least squares objectivefunction defined in equation (37) thenDpE only detects the edges of a perturbationif the signal wavelength is too short. The interior of the perturbation remains ’in-visible’ (Gauthier et al. 1986). However, choosing the signal wavelength severaltimes longer than the dimension of the perturbation, causes the perturbation to actas a diffractor and therefore it maps into the derivative as one single but relativelybroad peak. It can be concluded that such a waveform inversion relies on a suffi-ciently precise background model in order to guarantee that the remaining param-eter differences between the true Earth model and the numerical model merely actas scatterers. The background model can be obtained by a conventional traveltimetomography. Alternatively one could start with very long wavelengths which thensuccessively decrease in the course of the waveform inversion.The extent to which the lack of traveltime information in the sensitivity maps tothe properties of the derivative of an objective functionE depends on the definitionof E itself. Luo & Schuster (1991) explicitly introduced traveltime differences into
20
the objective function. Their approach overcomes the problem that only the edgesof perturbations are detected but it also results in a lower resolution.Pratt et al. (1998) proposed to use also the second derivative of the least squaresobjective function to solve the minimisation problem. They demonstrated thatDpE
multiplied by the inverse Hessian images a perturbation better thanDpE alone.However, the calculation and inversion of the second derivative is computationallytoo expensive for sufficiently realistic Earth models.The problem of missing traveltime information could alternatively be circumventedby using Monte Carlo methods in order to minimise the objective function. Thisapproach relies on the evaluation of a large number of parameter space elementsp ∈ P and it therefore becomes impractical if the solution of the forward problemis as time consuming as it is in seismology.It is important to note that sensitivities and first derivatives in general are physicallymeaningful only if the problem is either linear or if it can be reasonably linearized.The latter requires that the reference model be sufficiently close to the true modelso that parameter differences only act as scatterers. If significant non-linearities re-main, the first derivatives ofE andu only bear insufficient information.Dipolar sources are of particular interest in seismology because they allow us torepresent the sources of most tectonic earthquakes observed at great distances fromthe source region. By considering the elastic wave equation in an unbounded homo-geneous medium we found that the sensitivity kernels for each parameter (densityρ and the two Lame parametersµ andλ), decompose into four summands to whichwe referred as P→P, P→S, S→P and S→S kernels. The X→Y kernels, where Xand Y are either P or S, may be interpreted in terms of secondary Y-waves set off bya primary X-wave incident on a parameter perturbation. Generally, the sensitivitykernels can be represented by a product of two factors. In the case of the sensitivitykernels for density, the two factors correspond to the radiation patterns of the for-ward field and the adjoint field, respectively. The factors involved in the sensitivitykernels forλ andµ are more complex than the normal radiation patterns of elasticwaves.All sensitivity kernels exhibit singularities at the source and receiver and are smallin the region between them. This together with the fact that the first derivative onlyaccounts for scattering implies that waveform residuals of the direct waves (P orS) will almost exclusively map into the source and receiver regions. Therefore oneshould exclude either the direct waves or the source and receiver regions from theinversion procedure.Some of the sensitivity kernels are almost restricted to regions behind the sourceor receiver. Consequently, one may wish to exclude these unfavourable source-receiver configurations from the inversion. This requires information about boththe source orientation and the waveform that one attempts to reconstruct. Goodquality information about the source orientation is usually available. It is howeverpractically impossible to identify a waveform residual as being due to a particularX→Y conversion arising from parameter differences between the true Earth modeland the numerical model.In the same context we also found that the S→S kernel is larger than the S→P and
21
P→S kernels, which in turn are larger than the P→P kernels. Since this is true forany parameter perturbation, with the exception of perturbations inλ which do notinvolve S-waves, we conclude that waveform changes corresponding to the S ker-nels dominate those corresponding to the pure P kernels. This implies that S-waveresiduals, i.e., residuals with zero divergence, will dominate the misfit functionalor objective function. The implications of such an unbalanced use of waveforminformation on the actual inversion outcome certainly need further investigation.Generally it can be expected that variations inλ will be more difficult to detectthan variations inµ or ρ because they do not generate S-waves. One may correctfor this effect of S-wave dominance by including measurements of rotational mo-tion that allow to separate P- and S-wave motion (e.g. Nigbor, 1994; Pancha et al.,2000; Igel et al., 2005). Alternatively, one could only consider the waveform of thedirect P wave, therefore minimising the influence of S-waves. This would unfortu-nately eliminate the major advantage of waveform inversion, namely the possibilityof accounting for a maximum of information contained in a seismogram.Most of the results obtained in the course of this study are based on very simplemodels that are not realistic. But still, since the deduced phenomena have funda-mental character one can assume that they will also be present in more complicatedscenarios.An adjoint method based waveform inversion should also be viewed in the contextof seismic inversions based on ray theory and truncated normal mode expansions(e.g. Masters et al., 1996; Grand, 1997; Megnin & Romanowicz, 2000; Kennett& Gorbatov, 2004). Seismic imaging based on ray theory relies on a dense sam-pling of the regions of interest because ray traveltime sensitivities are in theoryspatially restricted to the ray. Therefore, a localised traveltime anomaly will not bevisible if there is no ray that penetrates it. However, sensitivity off the ray path canbe artificially introduced through a parametrization of the model with sufficientlyextended basis functions and through regularization. This produces smoothed in-homogeneities that may influence ray paths that are unaffected by a rough andspatially less extended inhomogeneity. Normal modes map local effects into globalones. This is an advantage because the normal mode waveforms, obtained by a fi-nite summation over global basis functions contain information about every pointinside the Earth. This, however, affects the resolution negatively. If based on theadjoint method, waveform inversion is in that sense between the two extremes ofray theory and normal modes. Depending on the particular sensitivity distribution,a parameter perturbation can be observed even if there is no ray passing through it.But still, this parameter perturbation will not have a global effect as it would if thesynthetic seismograms were computed via a truncated normal mode expansion. Thefact that the first derivative of the observed wavefield with respect to the parametersdoes not account for traveltime differences determines the possible applications ofan adjoint method based waveform inversion. It may for example be used to deter-mine the sharpness of discontinuities such as those caused by phase transitions inthe Earth’s mantle. Other regions where such a waveform inversion may be usefulinclude the thermal boundary layers, i.e., the crust and D”, where one may expectmore short wavelength heterogeneity than in the remaining mantle.
22
6 Conclusions
Departing from the definition of sensitivity functionals that allow us to express thesensitivity in terms of Green’s functions, we found that the first derivative of an ob-served wavefield with respect to the model parameters does not account for travel-time differences. The direct consequence is that a gradient-method based waveforminversion relies on an accurate background model. This background model may ei-ther be obtained through a conventional traveltime inversion or by using longerwavelengths. In the case of elastic wavefields set off by dipolar sources we saw thatthe surprisingly complex shape of waveform sensitivity kernels depends stronglyon both the source orientation and the observation direction. The fact that mostkernels vanish in the region between source and receiver implies that one shouldexclude either the direct waves or the source and receiver regions from a waveforminversion. The effect of S-wave residual dominance may significantly affect the in-version outcome, therefore posing the problem of exactly characterising differentwave types in a seismogram. This may in the future be accomplished by includingrotational measurements that allow a separation of P- and S-wave motion. Possibleapplications of the adjoint method in the context of waveform inversion includethe imaging of smaller scale structures that act as scatterers and which cannot beresolved by ray tomography or inversions based on truncated normal mode expan-sions. Moreover, it may be possible to determine the sharpness of discontinuitieswith increased accuracy.
Acknowledgements
The authors wish to thank Brian Kennett, Bernhard Forkmann and Steve Haney fordiscussions and constructive criticism.
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A Sensitivity kernels
We begin with the derivation of the sensitivity kernels for density. The kernelscorresponding to the parametersλ andµ can be derived in a similar way. Insertingthe expressions of Green’s functions in equations (46) and (47) into the densitysensitivity functional given in equation (45b) yields
Sρpqn(ξ0, τ0;h, ρ
′) =
∞∫−∞
t1∫t=t0
∫R3
∑X,Y
ρ′ aXFFpq · bY FF
n δ(t− τ − t0 − r(s)/cX
)δ(r(r)/cY + t− τ0
)h(τ) dτ dt d3x , (A.1)
where the summation∑
X,Y is over the four possible combinations ofX = P, SandY = P, S. Moreover,cP = α andcS = β. The vectorsaXFF
pq andbY FFn are
25
defined by
aPFFpq =
γ(s)p γ(s)
q
4πρα3r(s)
r(s)
r(s), aSFF
pq =γ(s)
q
4πρβ3r(s)
(ep −
r(s)
r(s)γ(s)
p
),
bPFFn =
γ(r)n
4πρα2r(r)
r(r)
r(r), bSFF
n =1
4πρβ2r(r)
(en −
r(r)
r(r)γ(r)
n
). (A.2)
Integration with respect toτ gives
Sρpqn(ξ0, τ0;h, ρ
′) =
t1∫t=t0
∫R3
∑X,Y
ρ′ aXFFpq · bY FF
n h(t− t0 − r(s)/cX
)δ(r(r)/cY + t− τ0
)dt d3x . (A.3)
The remaining integral is non-zero only if the observation timeτ0 satisfies the con-dition
τ0 ∈ (t0 + r(r)/cY , t1 + r(r)/cY ) . (A.4)Note that we also requireτ0 ∈ [t0, t1] which means that the observation time mustbe in the observation interval. Assuming that (A.4) is satisfied, we obtain
Sρpqn(ξ0, τ0;h, ρ
′) = −∫R3
∑X,Y
ρ′ aXFFpq ·bY FF
n
...h(τ0 − t0 − r(s)/cX − r(r)/cY
)d3x .
(A.5)The causality of the source time functionh implies thatSρ
pqn is non-zero only if
τ0 ≥ t0 + r(s)/cX + r(r)/cY . (A.6)
The integrations in the case of the sensitivity functional forµ are identical to thoseoutlined for the density sensitivity functional. It therefore suffices to derive thekernels themselves. For the gradients of the different Green’s functions we find
∇gPFFpq (x0, t0;x, t)
FF= −APFF
pq δ(t− t0 − r(s)/α) , (A.7a)
∇gSFFpq (x0, t0;x, t)
FF= −ASFF
pq δ(t− t0 − r(s)/β) , (A.7b)
∇g∗PFFn (x0, τ0;x, t)
FF= BPFF
n δ(r(r)/α+ t− τ0) , (A.7c)
∇g∗SFFn (x0, τ0;x, t)
FF= BSFF
n δ(r(r)/β + t− τ0) , (A.7d)
whereFF indicates that all near field terms generated through the differentiationhave been neglected. The cartesian components of the second-order tensorsAXFF
pq
andBY FFn are given by
(APFF
pq
)ij
=γ(s)
p γ(s)q γ
(s)i γ
(s)j
4πρα4r(s),
(ASFF
pq
)ij
=γ(s)
q γ(s)i
4πρβ4r(s)
(δjp − γ
(s)j γ(s)
p
),
(BPFF
n
)ij
=γ
(r)j γ
(r)i γ(r)
n
4πρα3r(r),
(BSFF
n
)ij
=γ
(r)i
4πρβ3r(r)
(δjn − γ
(r)j γ(r)
n
).
(A.8)
26
These tensors must be multiplied according to equations (45e) and (53) in order togive the sensitivity kernels. The sensitivity functional forλ is substantially simplerthan the ones forρ andµ due to the fact that the divergence of all S-wave Green’sfunctions has no far field component. For the P-wave Green’s functions we find
∇ · gPFFpq (x0, t0;x, t)
FF= −aFF
pq δ(t− t0 − r(s)/α) , (A.9a)
∇ · g∗PFFn (x0, τ0;x, t)
FF= bFF
n δ(r(r)/α+ t− τ0) . (A.9b)
The two scalarsaFFpq andbFF
n are
aFFpq =
γ(s)p γ(s)
q
4πρα4r(s), bFF
n =γ(r)
n
4πρα3r(r). (A.10)
27