Scattering Control for the Wave Equation with Unknown Wave Speed
Peter Caday, Maarten V. de Hoop, Vitaly Katsnelson, Gunther Uhlmann
SScattering Control for the Wave Equation with UnknownWave Speed
Peter Caday * , Maarten V. de Hoop † , Vitaly Katsnelson ‡ , and Gunther Uhlmann (cid:107) May 23, 2018
Abstract
Consider the acoustic wave equation with unknown wave speed c , not necessarily smooth. We proposeand study an iterative control procedure that erases the history of a wave field up to a given depth in amedium, without any knowledge of c . In the context of seismic or ultrasound imaging, this can be viewedas removing multiple reflections from normal-directed wavefronts. Consider the acoustic wave equation with an unknown wave speed c , not necessarily smooth, on a finite orinfinite domain Ω Ă R n . Assume that we can probe our domain Ω with arbitrary Cauchy data outside of Ω ,and measure the reflected waves outside Ω for sufficiently large time. The inverse problem is to deduce c from these reflection data, and this is the basis for many wave-based imaging methods, including seismicand ultrasound imaging.Toward this goal, we will define and study a time reversal-type iterative process, the scattering control series .We were inspired by the work of Rose [14] in one dimension, who developed a “single-sided autofocusing”procedure and identified it as Volterra iteration for the classical Marchenko equation. The Marchenkoequation solves the inverse problem for the one-dimensional acoustic wave equation , recovering c on ahalf-line from measurements made on the boundary. In the course of our research, it became evident thatthe new procedure is quite closely linked to boundary control problems [2, 8], and has similar properties toBingham et al.’s iterative time-reversal control procedure [3].In essence, scattering control allows us to isolate the deepest portion of a wave field generated by givenCauchy data— behavior we demonstrate with both an exact and microlocal (asymptotically high-frequency)analysis. Along the way we present several applications of scattering control, including the removal ofmultiple reflections and the measurement of energy content of a wave field at a particular depth in Ω . In afuture paper, we anticipate illustrating how to locate discontinuities in c and recover c itself.In the mathematical literature, the inverse problem’s data are typically given on the boundary of Ω , interms of the Dirichlet-to-Neumann map or its inverse. We find that the Cauchy data-based reflection mapallows us a much cleaner analysis. It is not hard to see (cf. Proposition 2.7) that the Dirichlet-to-Neumannmap determines the Cauchy data reflection map, so no extra information is needed.We start with an informal, graphical introduction to the problem. Section 2 defines the scattering controlseries rigorously and provides an exact analysis of its behavior and convergence properties. Section 3pursues the same questions from a microlocal perspective. The discrepancy that arises between the exactand microlocal analyses allows us to provide more insight on convergence in Section 4. Section 5 concludesby connecting our work to that of Rose and Marchenko. * [email protected] † [email protected] ‡ [email protected] (cid:107) [email protected] * † ‡ Department of Computational and Applied Mathematics, Rice University. (cid:107)
Department of Mathematics, University of Washington and Institute for Advanced Study, Hong Kong University of Science andTechnology. More precisely, the Marchenko equation treats the constant-speed wave equation with potential, to which the one-dimensionalacoustic wave equation can be reduced by a change of coordinates. a r X i v : . [ m a t h . A P ] M a y (a) Wave field generated by Cauchy data h T tT x (b)
Wave field with trailing pulse added to initialdata)
Figure 1.1: (a) A domain Ω (shaded) with unknown wave speed c is probed by exterior Cauchy data h . Twodiscontinuities in c (dashed) scatter the incoming wave. (b) An appropriate trailing pulse added to h suppressesmultiple reflections. Before defining the scattering control equation and series, we begin by motivating our problem with agraphical example. In Figure 1.1, the domain is Ω “ t x ą u Ă R , with a piecewise constant wave speed c having two discontinuities. We extend c to all of R , but assume it is known only outside Ω . Now considerthe solution of the acoustic wave equation on R for time t P r , T s , with rightward-traveling Cauchy data h supported outside Ω . The initial wave scatters from the discontinuities in c , producing an infinite sequenceof reflections (Figure 1.1(a)).In imaging, one attempts to recover c or some proxy for it. In many imaging algorithms currently in use,only waves having undergone a single reflection (so-called primary reflections ) are typically desired, while theremaining multiple reflections only complicate the interpretation of the data. As a result, much research inseismic imaging has been directed toward removing or attenuating multiple reflections.For the problem at hand, it is plausible (and can be proven) that by adding a proper control, or trailingpulse to the initial data, the multiple reflections may be suppressed, at the cost of a harmless additionaloutgoing pulse (Figure 1.1(b)). If c were known inside the domain (cf. §3.4), an appropriate control maybe constructed microlocally under some geometric conditions. The issue, of course, is to find the controlknowing only the reflection response of Ω .Rather than attacking the multiple reflection suppression problem, however, we consider a relatedproblem obtained by focusing on the interior, rather than exterior, of Ω . Returning to Figure (b), we notethat the wave field rightmost portion of the medium contains a single, purely transmitted wave, whichwe call the direct transmission of the initial data h . Slightly more precisely, the wave field inside Ω at time T is generated exactly by the direct transmission at time T . The control has therefore isolated the directtransmission; our problem is to find such a control for a given h using only information available outside Ω . At its heart, the direct transmission is a geometric optics construction, and is valid only in the high-frequencylimit where geometric optics holds. Consequently, the directly transmitted wave field can be isolated onlymicrolocally (modulo smooth functions). We will consider the geometric optics viewpoint later, but initiallyavoid a microlocal approach, as follows. Informally, suppose h creates a wave that enters Ω at time 0,travelling normal to the boundary. At a later time T , the directly transmitted wave may be singled out fromall others by its distance from the boundary: namely, T (as long as it has not crossed the cut locus). By2 Ω h B Θ B Θ T Support of wave field at time T Almost direct transmissionDirectly transmitted rayScattered rays
TTT
Figure 1.2:
Almost direct transmission of initial data h at time T ą . ∂ Ω ∂ Θ T ∂ Θ h Figure 1.3:
Shrinking the support of the initial data h to a point. The dashed line indicates the normal geodesicfrom that point; the support of the almost direct transmission shrinks to a point on the geodesic. distance we mean the travel time distance, which for c smooth is Riemannian distance in the metric c ´ dx .With this in mind, given Cauchy data h supported just outside Ω we substitute for the direct transmissionthe almost direct transmission , the part of the wave field of h at time T of depth at least T . More precisely, let Θ be a domain containing Ω and supp h ; then let Θ T Ă Θ be the set of points in Θ greater than distance T from the boundary. The almost direct transmission of initial data h at time T is the restriction to Θ T of itswave field at t “ T (Figure 1.2).The nonzero volume of Θ z Ω means that some multiply reflected rays may still reach Θ T . Hence, we havein mind taking a limit as Θ Ñ Ω and the support of h approaches a point on B Ω . In this limit, the support ofthe almost direct transmission converges to a point along the normal directly-transmitted ray, for sufficientlysmall T (at least in the absence of caustics and before reaching the cut locus); see Figure 1.3. We set up the problem and our notation in §2.1, then introduce the scattering control procedure in §2.2,where we study its behavior and convergence properties. The final result, expressed in Corollary 2.4, isthat scattering control recovers the almost direct transmission’s wave field outside Θ , modulo harmonicextensions. In §2.3, we apply this to recover the energy (with a harmonic extension) and kinetic energy ofthis portion of the wave field. Proofs for the results in these sections follow in §2.4. Let Ω Ď R n be a Lipschitz domain, and let c be a wave speed satisfying c, c ´ P L p R n q .Initially, the sole extra restriction we impose on c is that it satisfy a certain form of unique continuation.More precisely, assume there is a Lipschitz distance function d p x, y q such that any u P C p R , H p R n qq satisfying either: 3 u, B t u “ for t “ and d p x, x q ă T (finite speed of propagation)• u “ on a neighborhood of r´ T, T s ˆ t x u (unique continuation)is also zero on the light diamond D p x , T q “ tp t, x q | d p x, x q ă T ´ | t | u , if pB t ´ c ∆ q u “ on a neighborhood of D p x , T q , for any x P R n , T ą .While the set of wavespeeds with this property has not been settled in general, several large classes of c are eligible, stemming from the well-known work of Tataru [21]. Originally known for smooth soundspeeds [16, Theorem 4], Stefanov and Uhlmann later extended this to piecewise smooth speeds with conormalsingularities [17, Theorem 6.1], and Kirpichnikova and Kurylev to a class of piecewise smooth speeds in acertain kind of polyhedral domain [11, §5.1]. The corresponding travel time d p x, y q is the infimum of thelengths of all C curves γ p s q connecting x and y , measured in the metric c ´ dx , such that γ ´ p singsupp c q has measure zero. Next, let us set up the geometry of our problem. We will probe Ω with Cauchy data (an initial pulse )concentrated close to Ω , in some Lipschitz domain Θ Ą Ω . We will add to this initial pulse a Cauchy datacontrol (a tail ) supported outside Θ , whose role is to remove multiple reflections up to a certain depth,controlled by a time parameter T P p , diam Ω q . This will require us to consider controls supported in aLipschitz neighborhood Υ of Θ that satisfies d pB Υ , Θ q ą T and is otherwise arbitrary.While we are interested in what occurs inside Ω , the initial pulse region Θ will actually play a larger rolein the analysis. First, define the depth d ˚ Θ p x q of a point x inside Θ : d ˚ Θ p x q “ ` d p x, B Θ q , x P Θ , ´ d p x, B Θ q , x R Θ . (2.1)Larger values of d ˚ Θ are therefore deeper inside Θ . For each t , define the open sets Θ t “ t x P Υ | d ˚ Θ p x q ą t u , Θ ‹ t “ t x P Υ | d ˚ Θ p x q ă t u . (2.2)As in (2.2) above, we use a superscript ‹ to indicate sets and function spaces lying outside, rather than inside,some region. Let ˜ C be the space of Cauchy data of interest: ˜ C “ H p Υ q ‘ L p Υ q , (2.3)considered as a Hilbert space with the energy inner product @ p f , f q , p g , g q D “ ż Υ ` ∇ f p x q ¨ ∇ g p x q ` c ´ f p x q g p x q ˘ dx. (2.4)Within ˜ C define the subspaces of Cauchy data supported inside and outside Θ t : H t “ H p Θ t q ‘ L p Θ t q , H “ H , ˜ H ‹ t “ H p Θ ‹ t q ‘ L p Θ ‹ t q , ˜ H ‹ “ ˜ H ‹ . (2.5) We tacitly assume throughout that Θ t , Θ ‹ t are Lipschitz. h “ p h , h q P ˜ C in a subset W Ď R n : E W p h q “ ż W ´ |∇ h | ` c ´ | h | ¯ dx, KE W p h q “ ż W c ´ | h | dx. (2.6)Next, define F to be the solution operator [13] for the acoustic wave initial value problem: F : H p R n q ‘ L p R n q Ñ C p R , H p R n qq , F p h , h q “ u s.t. $’&’% pB t ´ c ∆ q u “ ,u (cid:12)(cid:12) t “ “ h , B t u (cid:12)(cid:12) t “ “ h . (2.7)Let R s propagate Cauchy data at time t “ to Cauchy data at t “ s : R s “ p F, B t F q ˇˇˇ t “ s : H p R n q ‘ L p R n q Ñ H p R n q ‘ L p R n q . (2.8)Now combine R s with a time-reversal operator ν : ˜ C Ñ ˜ C , defining for a given TR “ ν ˝ R T , ν : p f , f q ÞÑ p f , ´ f q . (2.9)In our problem, only waves interacting with p Ω , c q in time T are of interest. Consequently, let us ignoreCauchy data not interacting with Θ , as follows.Let G “ ˜ H ‹ X ` R T p H p R n z Θ q ‘ L p R n z Θ qq ˘ be the space of Cauchy data in ˜ C whose wave fields vanishon Θ at t “ and t “ T . Let C be its orthogonal complement inside ˜ C , and H ‹ t its orthogonal complementinside ˜ H ‹ t . With this definition, R maps C to itself isometrically. Θ t The final ingredients needed for the iterative scheme are restrictions of Cauchy data inside and outside Θ .While a hard cutoff is natural, it is not a bounded operator in energy space: a jump at B Θ will have infiniteenergy. The natural replacements are Hilbert space projections. More generally, we consider projectionsinside and outside Θ t .Let π t , π ‹ t be the orthogonal projections of C onto H t , H ‹ t respectively; let π t “ ´ π ‹ t . As usual, write π “ π , π ‹ “ π ‹ . The complementary projection I ´ π t ´ π ‹ t is the orthogonal projection onto I t , theorthogonal complement to H t ‘ H ‹ t in C . It may be described by the following lemma, which is in essencethe Dirichlet principle. Lemma 2.1. I t consists of all functions of the form p i , q , where i P H p Υ q is harmonic in Υ zB Θ t . Lemma 2.1 provides two useful pieces of information. First, I “ I is independent of c . Secondly, we canidentify the behavior of the projections π t , π ‹ t . Inside Θ t the projection π t h equals h , while outside Θ t , itagrees with the I t component of h , which is the harmonic extension of h | B Θ t to Υ (with zero trace on B Υ ).Similarly, π ‹ t h is zero on Θ t , and outside Θ t equals h with this harmonic extension subtracted.It will be useful to have a name for the behavior of π t h , and so we define the notion of stationaryharmonicity: Definition.
Cauchy data p h , h q are stationary harmonic on W Ď R n if h | W is harmonic and h | W “ . Suppose we have Cauchy data h P H . We can probe Ω with h and observe Rh outside Ω . In particular,the reflected data π ‹ R can be measured, and from these data, we would like to procure information about c inside Ω . However, multiple scattering as waves travel into and out of Ω makes π ‹ Rh difficult to interpret.In this section, we construct a control in H ‹ that eliminates multiple scattering in the wave field of h upto a depth T inside Θ . More specifically, consider the almost direct transmission of h :5 efinition. The almost direct transmission of h P H at time T is the restriction R T h | Θ T .Ideally, we would like to recover (indirectly) this restricted wave field. If considered as Cauchy data onthe ambient space Υ , the almost direct transmission has infinite energy in general due to the sharp cutoff atthe boundary of Θ T . As a workaround, consider the almost direct transmission’s minimal-energy extensionto Υ . This involves a harmonic extension of the first component of Cauchy data: Definition.
The harmonic almost direct transmission of h at time T is h DT “ h DT p h , T q “ π T R T h . (2.10)By Lemma 2.1, h DT is equal to R T h inside Θ T ; outside Θ T , its first component is extended harmonicallyfrom B Θ T , while the second component is extended by zero. Our major tool is a Neumann series, the scattering control series h “ ÿ i “ p π ‹ Rπ ‹ R q i h , (2.11)formally solving the scattering control equation p I ´ π ‹ Rπ ‹ R q h “ h . (2.12)The series in general does not converge in C ; but it does converge in an appropriate weighted space, aswe show in Theorem 2.3. Applying π to (2.11), we see that h consists of h plus a control in H ‹ . Our firsttheorem characterizes the behavior of the series. Theorem 2.2.
Let h P H and T P p , diam Θ q . Then isolating the deepest part of the wave field of h is equivalentto summing the scattering control series: p I ´ π ‹ Rπ ‹ R q h “ h ðñ R ´ T πR T h “ h DT and h P h ` H ‹ . (2.13) Above, R ´ T πR T h may also be replaced by R ´ s π T ´ s R T ` s h for any s P r , T s .Such an h , if it exists, is unique in C . As for the harmonic extension in h DT , it is equal to πR T h outside Θ : h DT ˇˇ Θ ‹ “ j ˇˇ Θ ‹ , where πR T h “ p j , j q , (2.14) and is bounded: E Θ ‹ T p h DT q ď C (cid:107) h (cid:107) (2.15) for some C “ C p c, T q independent of h . Equation (2.13) tells us that the wave field created by h inside Θ at t “ T is entirely due to the harmonicalmost direct transmission at t “ T (Figure 2.1). More generally, the wave field of h agrees with that of h DT on its domain of influence. This is not true of h ’s wave field, where other waves, including multiplereflections, will pollute the wave field at time T . It follows that the tail h ´ h enters Ω and carries all ofthe scattered energy of h out with it. We will see this from an energy standpoint in Section 2.3 and from amicrolocal (geometric optics) standpoint in Section 3.The question now is to study whether the Neumann series (2.11) converges at all. Since R is an isometryand π ‹ a projection, we have (cid:107) π ‹ Rπ ‹ R (cid:107) ď . From our later spectral characterization, we know that (cid:107) π ‹ Rh (cid:107) ă (cid:107) h (cid:107) , strictly, for all h P H ‹ . This is also true for a completely trivial reason: we eliminated G whenconstructing C . What hinders convergence is that (cid:107) h (cid:107) ´ (cid:107) π ‹ Rh (cid:107) might be arbitrarily small; in other words,almost all the energy could be reflected off Θ . Note that if the series fails to converge, no other finite energycontrol in H ‹ can isolate the harmonic almost direct transmission of h ; see Proposition 2.5.6 d ∗ Θ ( x ) h T − T Th ∞ − h T h DT equal towave fieldof h DT scattered initialpulse and control ∂ Θ0 ·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ control initial pulse Figure 2.1:
Illustration of the wave field generated by scattering control, as given by Theorem 2.2.
In the next theorem, we investigate convergence via the spectral theorem. It turns out that the onlyproblem is outside Θ ; inside Θ the partial sums’ wave fields at t “ T do converge, and their energies arein fact monotonically decreasing. We will also demonstrate that the Neumann series converges in H for adense set of h , and identify a larger space in which the Neumann series converges for any h .For the statement of the theorem, define J to be the following space of Cauchy data, which, roughlyspeaking, remains completely inside or completely outside Θ in time T : J “ ` H X R p H q ˘ ‘ ` H ‹ X R p H ‹ q ˘ . (2.16)Let χ : C Ñ J be the orthogonal projection onto J . Theorem 2.3.
With h , T as in Theorem 2.2, define the partial sums h k “ k ÿ i “ p π ‹ Rπ ‹ R q i h . (2.17) Then the deepest part of the wave field can be (indirectly) recovered from t h k u regardless of convergence of the scatteringcontrol series: lim k Ñ8 R ´ T πR T h k “ R T χh “ h DT , (cid:107) πRh k (cid:107) Œ (cid:107) h DT (cid:107) . (2.18) The set of h for which the scattering control series converges in C , Q “ (cid:32) h P H ˇˇ p I ´ π ‹ Rπ ‹ R q ´ h P C ( , (2.19) is dense in H . For all h P H , the partial sum tails h k ´ h converge in a weighted space that can be formally writtenas I ? I ´ N p ´ χ q C , N “ πRπ ` π ‹ Rπ ‹ . (2.20)As an immediate corollary of (2.18), we recover in the limit the wave field generated by the harmonicalmost direct transmission outside Θ , using only observable data. Corollary 2.4.
Let F DT p t, x q “ p F h DT qp t ´ T, x q be the harmonic almost direct transmission’s wave field. Then p F h k qp t, x q ´ p F π ‹ R T h k qp t ´ T, x q Ñ F DT p t, x q as k Ñ 8 , (2.21) the convergence being H in space, uniformly in t .
7e end this section with three small propositions. The first states that the scattering control equation hasno solution if the Neumann series diverges.
Proposition 2.5.
Let h , T be as in Theorem 2.2, and suppose p I ´ π ‹ Rπ ‹ R q k “ h for some k P H ˚ . Then thescattering control series (2.11) converges. The second proposition characterizes the space H ‹ containing the Cauchy data controls. Essentially, eachcontrol is supported in a T -neighborhood of Θ and its wave field is contained in this neighborhood for t P r , T s , up to harmonic functions. Proposition 2.6.
The control space H ‹ consists of Cauchy data supported outside Θ whose wave fields are stationaryharmonic outside a T -neighborhood of Θ at t “ , T : H ‹ “ ! h P ˜ C ˇˇˇ π ‹´ T h “ π ‹´ T R T h “ πh “ ) . (2.22)The third proposition shows that our reflection data (the Cauchy solution operator F , restricted to theexterior of Ω ) is determined by the Dirichlet-to-Neumann map, which is the data usually assumed givenin boundary control problems and the inverse problem. As a result, our method requires no additionalinformation, from a theoretical standpoint. Proposition 2.7.
Let c , c be L wave speeds on a C domain Ω Ď R n . Extend c , c to Ω ‹ “ R n z Ω by settingthem equal to some c P C p R n q .Define solution operators F , F corresponding to c , c as in (2.7) , and Dirichlet-to-Neumann maps Λ i : g ÞÑ B ν u (cid:12)(cid:12) R ˆB Ω , where $’&’% pB t ´ c i ∆ q u “ ,u (cid:12)(cid:12) R ˆB Ω “ g,u (cid:12)(cid:12) t “ “ B t u (cid:12)(cid:12) t “ “ . (2.23) If Λ “ Λ , then F h (cid:12)(cid:12) R ˆ Ω ‹ “ F h (cid:12)(cid:12) R ˆ Ω ‹ for all h P H p Ω ‹ q ‘ L p Ω ‹ q . As a direct application of the results in §2.2, we show how scattering control can recover the energy of theharmonic almost direct transmission using only data outside Ω , assuming supp h Ă Θ z Ω . If the Neumannseries converges to some h P C , we can recover the energy directly from h , but if not, Theorem 2.3 allowsus to recover the same quantities as a convergent limit involving the Neumann series’ partial sums. In aforthcoming paper we demonstrate how these energies may be used in inverse boundary value problems forthe wave equation that arise in imaging. Proposition 2.8.
Let h P H , T ą , and suppose p I ´ π ‹ Rπ ‹ R q h “ h . Then we can recover the harmonic almostdirect transmission’s energy from data observable on Θ ‹ Y supp h : E R n p h DT q “ E R n ` h ˘ ´ E R n ` π ‹ Rh ˘ . (2.24) We can also recover the kinetic energy of the almost direct transmission (with no harmonic extension) from dataobservable on Θ ‹ Y supp h : KE Θ T p R T h q “ x h , h ´ Rπ ‹ Rh ´ Rh y . (2.25) Proposition 2.9.
Let h P H and T ą , and h k as before. We can recover the energy of the harmonic almost directtransmission as a convergent limit involving data observable on Θ ‹ Y supp h : E R n p h DT q “ lim k Ñ8 r E R n p h k q ´ E R n p π ‹ Rh k qs . (2.26)8 T T T h F h ∞ = vF h ∞ = F h h DT d ∗ Θ ( x )0 Figure 2.2:
Finite speed of propagation applied twice to wave field v . Similarly, for the kinetic energy of the almost direct transmission, KE Θ T p R T h q “ lim k Ñ8 ” E p h k q ` E p h q ´ E p π ‹ Rπ ‹ Rh k q` x π ‹ Rh k , h k ´ Rπ ‹ Rh k y ´ x h , Rπ ‹ Rh k ` Rh k y ı . (2.27) Proof of Theorem 2.2.
The proof is mostly a simple application of unique continuation and finite speed ofpropagation.
Equation (2.13) ( ñ ) Let v p t, x q “ F R ´ T πR T h be the solution of the wave equation with Cauchy data πR T h at t “ T . We will often consider Cauchy data at a particular time, and so define v “ p v, B t v q .Applying ¯ π to the defining equation p I ´ π ‹ Rπ ‹ R q h “ h implies πh “ h ; also p π ‹ v qp , ¨q “ , since “ π ‹ h “ π ‹ p I ´ π ‹ R ´ T π ‹ R T q h “ π ‹ R ´ T πR T h “ p π ‹ v qp , ¨q . (2.28)Outside of Θ , then, v p , ¨q and v p T, ¨q are equal to their projections in I , and therefore are stationaryharmonic. Equivalently, B t v and B tt v are zero on Θ ‹ for t “ , T .Because c is time-independent, B t v is also a (distributional) solution to the wave equation. If B t v P C p R , H p R n qq , then Lemma 2.10 applied to B t v gives B t v p T, ¨q “ B tt v p T, ¨q “ on Θ ‹ T ; it follows that v p T, ¨q isstationary harmonic on Θ ‹ T . For the general case, choose a sequence of mollifiers ρ (cid:15) Ñ δ in E p R q and applyLemma 2.10 to ρ (cid:15) p t q ˚ v to obtain the same conclusion.By finite speed of propagation (FSP), ¯ π | s | R s ¯ π “ ¯ π | s | R s for any s P R . Applying this twice, we find that in Θ T at time T , the solution v is equal to h ’s wave field, which in turn is equal to h ’s wave field (Figure 2.2): π T v p T, ¨q “ π T R ´ T πR T h FSP “ π T R ´ T R T h “ π T R T h FSP “ π T R T ¯ πh “ π T R T h def “ h DT . (2.29)However, since v p T, ¨q is stationary harmonic on Θ ‹ T , we can remove the projection on the left-hand side: π T R ´ T πR T h “ R ´ T πR T h . This proves the forward direction of (2.13). More generally, it follows that π T ´ s R T ` s h “ v p T ` s, ¨q “ R s h DT for s P r , T s . Indeed, v p T ` s, ¨q “ R T ` s h on Θ T ´ s by finite speedof propagation, and using Lemma 2.10 as above implies v p T ` s, ¨q is stationary harmonic on Θ ‹ T ´ s for s P r , T s . 9 quation (2.14) As above, apply Lemma 2.10 to B t v . This implies that B t v | r , T sˆ Θ ‹ “ . Hence v is constantin time in Θ ‹ . At time T , we have v p T, ¨q “ π T R T h , and the pressure field v p T, ¨q is the harmonic extensionof the first component of R T h | B Θ T . At time T , v equals πR T h on Θ ‹ by construction, proving (2.14). Equation (2.13) ( ð ) Conversely, suppose R ´ T πR T h “ h DT . Let v p t, x q “ p F h DT qp t ´ T, x q be the wavefield generated by the harmonic almost direct transmission. Since v p T, ¨q is stationary harmonic in Θ ‹ T wehave pB t v qp T, ¨q “ there. Applying finite speed of propagation, pB t v qp , ¨q “ on Θ ‹ , so p π ‹ v qp , ¨q “ .Because R ´ T πR T h “ h DT , the solution v is equal to p F πR T h qp t ´ T, x q , the wave field generatedby πR T h . Hence π ‹ R ´ T πR T h “ , and we have p I ´ π ‹ Rπ ‹ R q h “ p I ´ π ‹ R p π ‹ ` π q R q h “ p I ´ π ‹ q h “ πh . (2.30)Therefore h is a solution of the scattering control equation for some initial pulse πh ; by hypothesis, thisinitial pulse is h . Uniqueness of h Since R is unitary and π is a projection, any g P C satisfies (cid:107) π ‹ Rπ ‹ Rg (cid:107) ď (cid:107) π ‹ Rg (cid:107) ď (cid:107) g (cid:107) . (2.31)Now, suppose that p I ´ π ‹ Rπ ‹ R q g “ for some g P C . As g “ π ‹ Rπ ‹ Rg no energy can be lost in eitherapplication of π ‹ , and both inequalities of (2.31) are in fact equalities. Hence πg and πR T g must be zero,implying g P G . But by construction G X C “ t u , establishing uniqueness.Conversely, any g P G satisfies g “ π ‹ Rπ ‹ Rg by finite speed of propagation, so in fact G “ ker p I ´ π ‹ Rπ ‹ R q . Equation (2.15) Finally, since i “ h DT | Θ ‹ “ π T R T h | Θ ‹ , it follows immediately that (cid:107) i (cid:107) ď (cid:107) π T R T h (cid:107) ď (cid:107) R T h (cid:107) “ (cid:107) h (cid:107) . (2.32)The proof is complete.In the proof of Theorem 2.2, we used the following corollary of finite speed of propagation and uniquecontinuation: Lemma 2.10.
Let u P C p R , H p R n qq be a solution of pB t ´ c ∆ q u “ such that u p , ¨q “ u p T, ¨q “ B t u p , ¨q “B t u p T, ¨q “ on Θ ‹ . Then u is zero on the set D “ tp t, x q | d ˚ Θ p x q ă T ´ | t ´ T | u . Proof.
By finite speed of propagation, u is zero on a neighborhood of r , T s ˆ Θ ´ T ´ δ for all δ ą , and thusby unique continuation, also zero on the union of open light diamonds centered at points in r , T s ˆ B Θ ´ T ´ δ .This includes r , T s ˆ Θ ´ T { ´ δ , and repeating the argument, we find that u “ on all open light diamondscentered at points in r , T s ˆ Θ ´ T { n ´ δ for all n P Z and δ ą . The union of these open light diamonds is D . Proof of Theorem 2.3.
The proof is via the spectral theorem, which will also shed further light on the behaviorof the Neumann series.First, note R “ ν ˝ R T is self-adjoint as well as unitary, since R ˚ “ R ˚ T ˝ ν ˚ “ R ´ T ˝ ν “ ν ˝ R T .Divide R into two self-adjoint parts, N and Z : N “ π ‹ Rπ ‹ ` πRπ, Z “ π ‹ Rπ ` πRπ ‹ . (2.33)In other words, thinking of im π ‹ “ H ‹ and im π “ H ‘ I as two halves of C , the operator N describeswave movement within one half, while Z describes movement from one half to the other. For any f P H the10dentity f “ R f “ p N ` Z q f ` p N Z ` ZN q f holds. If f P H ‹ or f P H ‘ I , then p N Z ` ZN q f is in theopposite half from f , so N Z ` ZN “ , and N ` Z “ I when the domain is restricted to either half.Applying the spectral theorem to N , identify C with L p X, µ q for some set X and measure µ , upon which N acts as a multiplication operator n p x q . As Z and N do not commute, Z has no special form with respect tothis spectral representation.Since (cid:107) N (cid:107) ď (cid:107) R (cid:107) “ , we have | n | ď . Split X into two sets X “ n ´ pt´ , uq ,X “ n ´ pp´ , qq “ X z X . (2.34)For h P L p X , µ q , (cid:107) N h (cid:107) “ ˆż X n | h | dµ ˙ { “ (cid:107) h (cid:107) “ (cid:107) Rh (cid:107) , (2.35)implying Zh “ . Conversely, if Zh “ , then (cid:107) N h (cid:107) “ (cid:107) h (cid:107) , implying n “ ˘ on supp h . In consequence, L p X , µ q “ ker Z “ J , and hence χ is multiplication by the characteristic function of X .Returning to the Neumann series, since p π ‹ q “ π ‹ , rewrite h k as h k ´ h “ k ´ ÿ i “ p π ‹ Rπ ‹ Rπ ‹ q i p π ‹ Rπ ‹ qp π ‹ R ¯ π q h “ k ´ ÿ i “ n i ` Zh “ n ´ n k ´ n Zh . (2.36)Turning to πRh k now, since Zn “ ´ nZ on im π ‹ Q n i Zh and Z “ ´ n , πRh k “ Z p h k ´ h q ` nh “ Zn ´ n k ´ n Zh ` nh “ ´ n ´ n k ´ n Z h ` nh “ n k ` h . (2.37) n k ` h converges pointwise, monotonically, as a function in L p X, µ q : p πRh k qp x q “ n k ` h p x q Ñ nh p x q , | n p x q | “ , | n p x q | ă . @ x P X. (2.38)The convergence holds not only pointwise but also in L p X, µ q by dominated convergence. Its limit functionis exactly nχh “ Rχh , the projection of Rh onto J , proving the first limit in (2.18). Also, as a consequenceof the monotonicity, (cid:107) πRh k (cid:107) Œ (cid:107) Rχh (cid:107) “ (cid:107) χh (cid:107) .Hence, while the Neumann series t h k u may diverge, the component of Rh k in H ‘ I (and therefore inside Θ ) converges and is actually decreasing in energy. Proof of (2.20) Starting from (2.36), we wish to commute Z and the powers of n . In the weighted space L p X , p ´ n q µ q , h k ´ h Ñ n ´ n Zh “ n ´ n Z p ´ χ q h “ ´ Z n ´ n p ´ χ q h . (2.39)The factor p ´ χ q is a projection away from the kernel of Z , where p ´ n q ´ blows up. We may insert itbecause J “ ker Z , and therefore Zχ “ . After doing so, the second equality holds because p ´ χ q h lies inthe inside half H ‘ I .Any j P H (or H ‹ ) satisfies (cid:107) j (cid:107) “ (cid:107) Rj (cid:107) “ (cid:107) Zj (cid:107) ` (cid:107) N j (cid:107) , so (cid:107) Zj (cid:107) “ ż X p ´ n q | j | dµ “ (cid:13)(cid:13)(cid:13) a ´ n j (cid:13)(cid:13)(cid:13) . (2.40)11pplying this relation to h k ´ h , (cid:107) h k ´ h (cid:107) “ (cid:13)(cid:13)(cid:13)(cid:13) n ´ n k ? ´ n p ´ χ q h (cid:13)(cid:13)(cid:13)(cid:13) . (2.41)Therefore, h k ´ h lies in the weighted space L p X , p ´ n q µ q , and, by dominated convergence, convergesto a function h ´ h P L p X , p ´ n q µ q . Formally, this latter space can be written p I ´ N q ´ { p ´ χ q C ,establishing (2.20). Density of Q Decompose X as the disjoint union of the family of sets X ´ “ n ´ pt´ , , uq ; X i “ n ´ pp´ ` ´ i ´ , ´ ` ´ i q Y p ´ ´ i , ´ ´ i ´ qq i “ , , ¨ ¨ ¨ . (2.42)Let h p i q “ h ¨ X ´ \¨¨¨\ X i , where A denotes the indicator function of A Ď X . Then h p i q Ñ h in L p X, µ q .Using the fact that Zn “ ´ nZ on H ‹ , as before the k th partial sum of the Neumann series for h p i q is h p i q k “ h p i q ` n ´ n k ´ n Zh p i q “ h p i q ´ Zn ´ n k ´ n p ´ χ q h p i q . (2.43)Since either n “ ˘ (so that ´ χ “ ) or | n | ă ´ ´ i ´ , the multiplier n ´ n k ´ n p ´ χ q is bounded in k andthe Neumann series converges in C . Hence h p i q P Q for all i , proving Q is dense. Proof of
Rχh “ h DT When h k converges in C , by Theorem 2.2 we have lim k Ñ8 R ´ T πR T h k “ h DT . (2.44)The left hand side is equal to Rχh ; hence for h P Q , Rχh “ h DT . (2.45)By the unitarity of R and (2.15), h ÞÑ h DT is a continuous map from H to C . The left-hand side is likewisecontinuous in h . So, since Q is dense in H , (2.45) holds for all h P H . This together with our earlier workestablishes (2.18). By the same argument, h DT “ lim k Ñ8 R ´ s π T ´ s R T ` s h k for any s P r , T s . Proof of Proposition 2.8.
Equation (2.24) follows directly from (2.13): E p h DT q “ E p R ´ T πR T h q “ E p πRh q “ E p Rh q ´ E p π ‹ Rh q “ E p h q ´ E p π ‹ Rh q . (2.46)For (2.25), let v p t, x q “ p F πR T h qp t ´ T, x q , as in the proof of Theorem 2.2. Subtract its time-reversalto get the solution w p t, x q “ v p t, x q ´ v p T ´ t, x q , and as before write v “ p v, B t v q , w “ p w, B t w q . Considerthe energy of w at t “ T . Now w p T, ¨q “ everywhere and B t w “ B t v “ on Θ ‹ T (as shown by the proof ofTheorem 2.2), so the only energy of w at time T is inside Θ T : E p w p T, ¨qq “ ż R n c ´ | B t w p T, ¨q | dx “ ż R n c ´ | B t v p T, ¨q | dx “ KE Θ T p v p T, ¨qq FSP “ KE Θ T p R T h q FSP “ KE Θ T p R T h q . (2.47)The last two equalities are by finite speed of propagation, as in (2.29). By conservation of energy, E p w p T, ¨qq “ E p w p T, ¨qq “ E p πRh ´ πRπRh q . (2.48)Expanding out the energy norm on the right hand side, KE Θ T p R T h q “ (cid:107) πRh (cid:107) ` (cid:107) πRπRh (cid:107) ´ x πRh , πRπRh y . (2.49)12sing πRπRh ` πRπ ‹ Rh “ πh “ h , and π ‹ RπRh “ , (cid:107) πRh (cid:107) “ (cid:107) Rh (cid:107) ´ (cid:107) π ‹ Rh (cid:107) “ (cid:107) h (cid:107) ´ (cid:107) π ‹ Rh (cid:107) ; (cid:107) πRπRh (cid:107) “ (cid:107) h ´ πRπ ‹ Rh (cid:107) “ (cid:107) h (cid:107) ` (cid:107) πRπ ‹ Rh (cid:107) ´ x h , πRπ ‹ Rh y“ (cid:107) h (cid:107) ` (cid:107) π ‹ Rh (cid:107) ´ (cid:107) π ‹ Rπ ‹ Rh (cid:107) ´ x h , Rπ ‹ Rh y ; x πRh , πRπRh y “ x Rh , RπRh y ´ x π ‹ Rh , π ‹ RπRh y“ x h , πRh y“ x h , Rh y . (2.50)Recalling π ‹ Rπ ‹ Rh “ h ´ h and simplifying yields (2.25). Proof of Proposition 2.9.
Proof of (2.26) The energy recovery formula follows directly from Theorem 2.3: lim k Ñ8 r E p h k q ´ E p π ‹ Rh k qs “ lim k Ñ8 (cid:107) Rh k (cid:107) ´ (cid:107) π ‹ Rh k (cid:107) “ lim k Ñ8 (cid:107) πRh k (cid:107) “ (cid:107) h DT (cid:107) . (2.51) Proof of (2.27) The proof is similar to (2.25), but with extra terms. By (2.47)–(2.50), h satisfies KE Θ T p R T h q “ E p πRh ´ πRπRh q (2.52) “ E p h q ` E p h q ´ E p π ‹ Rπ ‹ Rh q ´ x h , Rπ ‹ Rh ` Rh y . (2.53)For h k , we must modify the second equality as π ‹ RπRh k is no longer zero. Instead, write π ‹ RπRh k as π ‹ h k ´ π ‹ Rπ ‹ Rh k to obtain E p πRh k ´ πRπRh k q “ E p h k q ` E p h q ´ E p π ‹ Rπ ‹ Rh k q` x π ‹ Rh k , π ‹ h k ´ π ‹ Rπ ‹ Rh k y ´ x h , Rπ ‹ Rh k ` Rh k y . (2.54)The right-hand side is the quantity in the limit in (2.27). As k Ñ 8 , it converges to (2.53) by continuity aslong as h P Q ; hence its limit is KE Θ T p R T h q . This proves (2.27) when h P Q . Then, by continuity andthe density of Q , (2.27) must hold for all h P H .Interestingly, to obtain kinetic energy we used initial data lim k Ñ8 r πRh k ´ πRπRh k s “ Rχh ´ πχh “ p n ´ q χh , (2.55)equal to ´ times the projection of h onto L p n ´ pt´ uq , µ q . Proof of Lemma 2.1.
The proof is essentially that of the Dirichlet principle. First, while H “ H ‹ t ‘ I t ‘ H t , wenote that also (with tildes) ˜ H “ ˜ H ‹ t ‘ I t ‘ H t . (2.56)This is true simply because I t is orthogonal to G and hence to ˜ H ‹ t “ H ‹ t ‘ G .Now, for one direction of the proof, consider an arbitrary i “ p i , i q P I t . Since Θ t is Lipschitz, itsboundary has measure zero, so L p Υ q “ L p Θ ‹ t q ‘ L p Θ t q . Hence i must be zero.Let φ P H t be nonzero and a ą . Then (cid:107) i ` aφ (cid:107) “ (cid:107) i (cid:107) ` a (cid:107) φ (cid:107) ą (cid:107) i (cid:107) by orthogonality. Hence a “ is a local minimum of (cid:107) i ` aφ (cid:107) , and the derivative of this quantity with respect to a is zero at a “ :13 “ dda (cid:107) i ` aφ (cid:107) ˇˇˇˇ a “ “ x i, φ y “ ż Υ ∇ i ¨ ∇ φ . (2.57)Since i is weakly harmonic on Θ t , it is strongly harmonic; in the same way it is harmonic on Θ ‹ t .Conversely, if i P H p Υ q is harmonic on Υ zB Θ t , it is weakly harmonic, immediately implying p i , q isorthogonal to H t and H ‹ t . Proof of Proposition 2.5.
First, we have the equivalence p I ´ π ‹ Rπ ‹ R q h “ h ðñ p I ´ π ‹ Rπ ‹ R qp h ´ h q “ π ‹ Rπ ‹ Rh . (2.58)Since π ‹ Rπ ‹ is self-adjoint and (cid:107) π ‹ Rπ ‹ (cid:107) ď (cf. the proof of Theorem 2.3), it suffices to apply the followinglemma. Lemma 2.11.
Let A be a self-adjoint linear operator on a Hilbert space X with (cid:107) A (cid:107) ď . If x, y P X satisfy p I ´ A q y “ x , then the Neumann series ř k “ A k x converges to the minimal-norm solution y “ y ˚ to p I ´ A q y “ x .Proof. By the spectral theorem, X can be identified with L p W, µ q for some set W and measure µ , upon which A acts as a (real-valued) multiplication operator a p w q ; also (cid:107) A (cid:107) ď implies | a | ď for all w P W . If i p w q denotes the indicator function of a ´ p˘ q , then y “ y ˚ “ iy is the minimal-norm solution of p I ´ A q y “ x .Let y n “ y n p w q “ ř nk “ a k x be the n th partial sum of the Neumann series; then y n p w q convergesmonotonically away from zero to yi for each w . Hence y n Ñ y ˚ in L p W, µ q . Proof of Proposition 2.6.
Our first task is to characterize G , the space of functions staying outside Θ in time T . We make a guess G for G and show that the two are equal by unique continuation, using Lemma 2.10.After identifying G , it will be easy to identify H ‹ , its complement in ˜ H ‹ .First, define G “ H p Θ ‹´ T q ‘ L p Θ ‹´ T q , G “ G ` R T G . (2.59)By finite speed of propagation, G , R T G Ď G , so G Ď G . We want to show that in fact G “ G .Accordingly, suppose g P G and g K G .Having g K G implies π ‹´ T g “ ; similarly g K R T G implies π ‹´ T Rg “ . That is, the wave field of g is stationary harmonic outside a T -neighborhood of Θ at t “ , T . As in the proof of Theorem 2.2, we canapply Lemma 2.10 to (a smoothed version of) B t F g to conclude that R T g is stationary harmonic outside a T -neighborhood of Θ at time T ; i.e., π ‹´ T R T g “ . (2.60)On the other hand, g P G implies that πg “ πRg “ ; the wave field of g is zero on Θ at t “ , T . ApplyingLemma 2.10, we can conclude that the wave field of g is zero on a T -neighborhood of Θ at time T ; i.e. π ´ T R T g “ . (2.61)Hence R T g “ π ‹´ T R T g ` π ´ T R T g “ ; we conclude that g “ , and therefore G “ G .Now, we can prove (2.22). H ‹ is the complement of G in ˜ H ‹ . For Cauchy data h P ˜ C , h P ˜ H ‹ ðñ πh “ , (2.62)and since G “ G , equations (2.60–2.61) imply h K G ðñ h K G and h K R G ðñ π ‹´ T h “ and π ‹´ T Rh “ . (2.63) Proof of Proposition 2.7.
Let h P H p Ω ‹ q ‘ L p Ω ‹ q , and let u “ F h be the solution with respect to c . Define u to be the solution of the IBVP (2.23) with boundary data u ˇˇ R ˆB Ω . Since c and c have identical Dirichlet-to-Neumann maps, it follows that B ν u ˇˇ R ˆB Ω “ B ν u ˇˇ R ˆB Ω . Therefore, u may be extended to R ˆ R n bysetting it equal to u outside Ω , and both u and B ν u will be continuous on R ˆ B Ω . Hence u satisfies thewave equation with respect to c inside and outside Ω , and satisfies the interface conditions at B Ω . Therefore,it is a solution of the c wave equation on all of R n [18, Theorem 2.7.3]. By uniqueness of the Cauchy problem, u “ F h , and by definition u “ u “ F h on Ω ‹ . 14 Microlocal analysis of scattering control
In this section, we turn from our exact analysis of scattering control to a study of its microlocal (high-frequency limit) behavior, allowing us to study reflections and transmissions of wavefronts naturally. Toaccomodate the microlocal analysis, we first narrow the setup somewhat, and consider a microlocally-friendly version of the scattering control equation in §3.1. Section 3.2 introduces a natural analogue of thealmost direct transmission, based on depths of singularities (covectors), rather than points.Just as before, isolating the microlocal almost direct transmission is sufficient for solving the microlocalscattering control equation (§3.3). If the wave speed c is known, it is not hard, as §3.4 shows, to constructsolutions assuming some natural geometric conditions. Our main result, Theorem 3.3, is that the scatteringcontrol iteration converges to a similar solution, to leading order in amplitude, under the same conditions.Finally, §3.6 discusses uniqueness for the microlocal scattering control equation. Proofs of the key resultsfollow in §3.7. Notation
Throughout, “ ” ” denotes equality modulo smooth functions or smoothing operators, and ˚ T ˚ M “ T ˚ M z ( M a manifold). A graph FIO is a Fourier integral operator associated with a canonical graph.Finally, for a set of covectors W Ď T ˚ M , let D W , E W denote the spaces of distributions with wavefront set in W . In this section, we begin by restricting Ω and c suitably in order to study reflection and transmission ofsingularities. We also adjust the scattering control equation slightly, replacing projections with smoothcutoffs, and employing a parametrix for wave propagation.Let Ω Ď R n be a smooth open submanifold, and c a piecewise smooth wave speed that is singular onlyon a set of disjoint, closed , connected, smooth hypersurfaces Γ i of Ω , called interfaces . Let Γ “ Ť Γ i ; let t Ω j u be the connected components of R n z Γ . Also assume each smooth piece of c extends smoothly to R n .The projections π , π ‹ arose quite naturally in the exact setting, taking the roles of cutoffs inside andoutside Θ . Because they introduce singularities along B Θ , it is natural to replace them by smooth cutoffs fora microlocal study. We will also separate the initial data h from the cutoff region. To accommodate bothaims, choose nested open sets Θ , Θ between Ω and Θ : Ω Ď Θ Ď Θ Ď Θ Ď Θ Ď Θ , (3.1)and smooth cutoffs σ, σ ‹ : R n Ñ r , s such that σ p x q “ , x P Θ , , x R Θ , supp σ “ Θ , (3.2) σ ‹ “ ´ σ, supp σ ‹ “ R n z Θ . (3.3)The sets Θ , Θ should be thought of as arbitrarily close to Θ ; we will write Θ “ R n z Θ .Finally, a standard parametrix ˜ R accounting for reflections and refractions will frequently replace the exactpropagator R , discussed at greater length in Appendix A. Most importantly, ˜ R includes microlocal cutoffsalong glancing rays, so that Rh ” ˜ Rh as long as WF p h q is disjoint from a set of covectors W Ă T ˚ p R n z Γ q producing near-glancing broken bicharacteristics.The object of study is now the microlocal scattering control equation p I ´ σ ‹ Rσ ‹ R q h ” h , (3.4) As usual, “smooth” means C throughout. If c is singular on some non-closed hypersurface Γ i , we may be able to “close up” Γ i in such a way that it does not intersect theother hypersurfaces. ΘΓ B Θ T h (a) Wavefront set of solution at time T ą T (b) Wavefront set of microlocal almost directtransmission (c)
Wavefront set of almost direct transmission
Figure 3.1:
Microlocal almost direct transmission. (a) The wavefront set of the solution with point source h includes reflected and refracted singularities due to an interface Γ . (b) The microlocal almost direct transmissiondoes not include the reflected singularities; their depth is less than T . (c) Wavefront set of the (non-microlocal)almost direct transmission, for comparison. and accompanying formal Neumann series h ” ÿ i “ p σ ‹ R q i h . (3.5)In general, the operator p σ ‹ R q preserves but does not improve Sobolev regularity, preventing us fromassigning any meaning to this infinite sum a priori . Instead, we will consider the limiting behavior of itspartial sums.
The almost direct transmission played a central role in the exact analysis of scattering control. We beginby studying its natural microlocal analogue. Intuitively, the microlocal almost direct transmission h MDT is themicrolocal restriction of the solution at time T to singularities in ˚ T ˚ Θ whose distance from the surface B T ˚ Θ is at least T (Figure 3.1). The distance here should be defined as the length of the shortest brokenbicharacteristic segment connecting a covector to the boundary (Figure 3.2). In general, our h MDT is notequivalent to the ideal direct transmission, which would contains only transmitted waves, but it may stillserve as a useful proxy.In the remainder of the section, we briefly define distance in the cotangent bundle, then use it to definethe microlocal almost direct transmission h MDT . Distance in the Cotangent Bundle
Let V “ R ˆ p R n z Γ q . For brevity, we shall simply say γ : p s ´ , s ` q Ñ ˚ T ˚ V ˘ is a bicharacteristic if it is a bicharacteristic for B t ´ c ∆ ; is unit speed , i.e., dt { ds “ on γ ; and is maximal , Were p σ ‹ R q to have negative Sobolev order, (3.5) may be interpreted as an asymptotic series. This situation occurs, for example,for c with C ,α or weaker singularities [9], in the absence of diving rays. Θ ξ Figure 3.2:
Depth of a singularity. The broken bicharacteristic segments joining covector ξ to the boundary areshown, projected to R n (solid); they reflect and refract at interfaces (dotted lines). The depth of ξ in T ˚ Θ is definedas the length of the shortest of these paths to the boundary (bold). i.e., cannot be extended. Here s ˘ may be infinite.A broken bicharacteristic γ : p s , s q Y p s , s q Y ¨ ¨ ¨ Y p s k ´ , s k q Ñ ˚ T ˚ V is a sequence of bicharacteristicsconnected by reflections and refractions obeying Snell’s law: for i “ , . . . , k ´ , γ p s ´ i q , γ p s ` i q P ˚ T ˚ pr , T s ˆ Γ q , p di Γ q ˚ γ p s ´ i q “ p di Γ q ˚ γ p s ` i q , (3.6)where i Γ : Γ ã Ñ Ω is inclusion. Since any broken bicharacteristic may be parameterized by time, we will oftenabuse notation and consider γ as a map from t P R into ˚ T ˚ p R n z Γ q .The distance of a covector ξ P ˚ T ˚ p R n z Γ q from the boundary of M Ď R n is d p ξ, B T ˚ M q “ min t | a ´ b | : γ p a q “ ξ, γ p b q P B T ˚ M u , (3.7)the minimum taken over broken bicharacteristics γ . Extend d p¨ , B T ˚ M q to all ξ P ˚ T ˚ R n by lower semiconti-nuity. In general, d will not be continuous at ˚ T ˚ p R ˆ Γ q . Depth is the same as distance, but with a sign indicating whether ξ is inside or outside M : d ˚ T ˚ M p ξ q “ ` d p ξ, B T ˚ M q , ξ P T ˚ M , ´ d p ξ, B T ˚ M q , otherwise . (3.8) Microlocal Almost Direct Transmission
Let p T ˚ M q t be the set of covectors of depth greater than t in amanifold M : p T ˚ M q t “ t ξ P T ˚ M | d ˚ T ˚ M p ξ q ą t u . (3.9)Figure 3.3 illustrates p T ˚ M q t in a simple case. Note p T ˚ M q t Ľ T ˚ p M t q in general, where M t is defined asin (2.2).A microlocal almost direct transmission of h at time T is a distribution h MDT satisfying h MDT ” R T h on p T ˚ Θ q T WF p h MDT q Ď p T ˚ Θ q T . (3.10)Essentially, h MDT is any sufficiently sharp microlocal cutoff of R T h outside p T ˚ Θ q T . Note that there is a gap G “ p T ˚ Θ q T zp T ˚ Θ q T in which we do not characterize h MDT ; the gap is needed in case WF p R T h q intersects Bp T ˚ Θ q T , since then the cutoff may not be infinitely sharp. The solutions of (3.10) form an equivalence classmodulo D G ` C p R n q , since any two choices of h MDT differ exactly by a distribution with wavefront set in G . With this equivalence class in mind, we denote by h MDT any solution of (3.10) and refer to it simply as the microlocal almost direct transmission. Note that WF p h MDT q Ă p T ˚ Θ q T Ă WF p h DT q Ă T ˚ p Θ T q . (3.11)It is natural to visualize h MDT with a depth diagram plotting the depths of the wave field’s singularities overtime (Figure 3.4). The depth of a singularity traveling along any broken bicharacteristic γ is a piecewiselinear function of time, with derivative ˘ almost everywhere, so a depth diagram consists of line segmentsof slope ˘ . Note that the depth of γ p t q is (up to sign) the shortest distance from γ p t q to the surface along any broken bicharacteristic, not only along γ . 17 Θ ∂ Θ T T Figure 3.3:
Example of a depth sublevel set p T ˚ Θ q T , with wave speed c “ . Each marked circle describes the unitcovectors based at its center point: those inside p T ˚ Θ q T are marked in black, those outside in white. Near theboundary, p T ˚ Θ q T contains only nearly horizontal covectors, while below Θ T it contains covectors in all directions,as the distance to the surface in any direction is greater than T . d ∗ T ∗ Θ ( ξ ) T Tth h MDT (a) Depths of singularities in wave field ∂ Θ h h MDT (b)
Wave field of h Figure 3.4:
Microlocal almost direct transmission: h MDT contains the singularities in R T h of depth at least T in T ˚ Θ . (a) Depth diagram; interfaces marked with small circles. (b) Projection onto R n ; interfaces dotted. α α α α ∂ Θ γ γ γ γ (a) Ray configuration with one interface d ∗ T ∗ Θ ( ξ ) t α α α t α (b) Depth diagram of bicharacteristics γ i Figure 3.5:
Depth discontinuity at interfaces. (a) Covectors α , α are closer to the boundary (via γ ) than α , whichcannot take this path. (b) Depths of the positive bicharacteristics γ i through these α i , meeting the interface at time t . A jump occurs at the interface along either broken bicharacteristic through α . ˚ T ˚ Θ p ξ q T Tth T h
MDT (a)
Wave field of h d ˚ T ˚ Θ p ξ q T Tth T R T h MDT h ´ h h MDT (b)
Wave field of h Figure 3.6:
Isolating h MDT . A singularity from h travels inward, reflecting and refracting from two interfaces(indicated by open circles). The multiply-reflected ray (dotted) will enter the domain of influence of h MDT (shaded).To prevent this, h must include an appropriate singularity to eliminate the multiply-reflected ray. The horizontalaxis is depth in the cotangent bundle. Remarks. • Along a broken bicharacteristic, d ˚ T ˚ Θ is often discontinuous at interfaces, as illustrated in Figure 3.5.To see why, consider a bicharacteristic γ encountering an interface; let γ , γ be the reflected andtransmitted bicharacteristics, and let γ be the opposite incoming bicharacteristic. In general, oneof the γ i , say γ , provides the shortest route from the interface to the boundary. Singularities along γ or γ can reach the boundary along γ , while those along γ cannot and must take a longer path.Consequently, a jump in depth occurs when passing from γ to either γ or γ .• Along a singly reflected bicharacteristic, depth does not switch from increasing to decreasing at themoment of reflection in general. Instead, depth will change from increasing to decreasing halfwayalong; compare the broken bicharacteristic γ Y γ in Figure 3.5.• Depth (and hence h MDT ) cannot intrinsically distinguish reflections from transmissions. This is possibleonly under geometric assumptions ensuring that reflected waves travel toward the boundary, andtransmitted waves travel away from it; e.g., Θ “ t x n ą u a halfspace, and c a function of x n alone. One of our earlier key facts, expressed in Theorem 2.2, is that solving the (exact) scattering control equation p I ´ π ‹ Rπ ‹ R q h “ h for h is equivalent to isolating the almost direct transmission: πR T h “ R T h DT (assuming h “ h on Θ ). In other words, the wave field of h at t “ T inside the domain Θ is exactly thealmost direct transmission’s wave field, undisrupted by any waves from shallower regions.Our main goal now is to consider the microlocal version of this equivalence: is solving the microlocalscattering control equation (3.4) equivalent to isolating h MDT ? As before, one direction is easy: if a tail h isfound that isolates h MDT (in the sense that R T h ” R T h MDT on Θ ) it is a solution of (3.4). The idea behindcrafting such an h we have seen already in Figure 1.1: h should include appropriate extra singularitiesthat ensure singularities in the wave field of h at depth less than T do not interfere with h MDT ’s wave field.Figure 3.6 illustrates the situation.
Lemma 3.1.
Let h P E p Θ z Γ q ‘ E p Θ z Γ q . Suppose h P E p R n z Γ q ‘ E p R n z Γ q isolates the microlocal almostdirect transmission, in the sense that h (cid:12)(cid:12) Θ ” h (cid:12)(cid:12) Θ and R T h (cid:12)(cid:12) Θ ” R T h MDT (cid:12)(cid:12) Θ . (3.12)19 hen h satisfies the microlocal scattering control equation, p I ´ σ ‹ Rσ ‹ R q h ” h . The same holds true with ˜ R replacing R .Proof. Let v p t, x q “ p F σR T h qp t ´ T, x q be the wave field generated by σR T h , and v “ p v, B t v q . Since WF p h MDT q Ď p T ˚ Θ q T , propagation of singularities limits the wavefront set of R T h MDT to T ˚ Θ , where thecutoff σ is identity. Hence v at time T agrees with R T h MDT . Moving to time T , we have v p T, ¨q ” f MDT ; bypropagation of singularities again, WF p v p , ¨qq Ď T ˚ Θ . In particular, σ ‹ RσRh “ σ ‹ v p , ¨q is smooth. Weconclude that σ ‹ Rσ ‹ Rh “ σ ‹ R p ´ σ q Rh ” σ ‹ h ´ ” h ´ h . (3.13)The same argument holds with the parametrix ˜ R in place of R .Just like Theorem 2.2, Lemma 3.1 assures us that solving the microlocal scattering control equation isnecessary for producing a tail h ´ h that isolates h MDT .The other direction of the problem (does a solution of the microlocal scattering control equation isolate h MDT ?) is a more subtle question, taken up in the following sections. Our overarching goal is to show that h MDT , like its non-microlocal version h DT , may be found by the Neumann-type iteration (3.5). We start byexplicitly constructing a Fourier integral operator A that isolates h MDT , given c . By Lemma 3.1 this FIO is amicrolocal inverse for I ´ σ ‹ Rσ ‹ R . Now, Neumann iteration also provides a (formal) microlocal inversefor this operator. The existence of A can be used to show that Neumann iteration isolates h MDT as well, in aprincipal symbol sense. This leads to the question of injectivity for I ´ σ ‹ Rσ ‹ R , explored in greater depth inSection 3.6. I ´ σ ‹ Rσ ‹ R In this section, we lay out conditions on Θ , c , h under which we can show the existence of an h isolating h MDT , and thereby I ´ σ ‹ Rσ ‹ R . The motivation for this relatively straightforward task is that it enables thestudy the convergence behavior of the microlocal Neumann iteration in the following section.We start by making a number of definitions; most of which are illustrated in Figure 3.7. Definition. (a) The forward and backward microlocal domains of influence D ` MDT , D ´ MDT are defined by: D ´ MDT “ tp t, η q P r , T s ˆ ˚ T ˚ R n | d ˚ T ˚ Θ p η q ą t u , D ` MDT “ tp t, η q P r T, T s ˆ ˚ T ˚ R n | d ˚ T ˚ Θ p η q ą T ´ t u . (3.14)By propagation of singularities, every η P WF p h MDT q is connected to some η P WF p h q by a brokenbicharacteristic inside D ´ MDT .(b) A returning bicharacteristic γ : p t ´ , t ` q Ñ ˚ T ˚ p R n z Γ q is one that leaves D ´ MDT before t “ T . Moreprecisely, γ p t q P D ´ MDT and lim t Ñ t γ p t q R D ´ MDT for some t , t P p t ´ , t ` s , t ă t .(c) Bicharacteristics γ , γ are connected if their union γ Y γ is a broken bicharacteristic. A bicharacteristic γ terminating in an interface may have one (totally reflected), or two (reflected and transmitted)connecting bicharacteristics there. If it has two, there exists an opposite bicharacteristic γ sharing γ ’sconnecting bicharacteristics.(d) A bicharacteristic γ : p t ´ , t ` q Ñ ˚ T ˚ p R n z Γ q is p˘q -escapable if either:i. it has escaped : γ is defined at t “ T ˘ T and γ p T ˘ T q R T ˚ Θ ,or recursively, after only finitely many recursions, either Note that for simplicity Figure 3.7 is not generic; in light of the remarks in §3.1, the behavior of d ˚ T ˚ Θ is typically much morecomplicated. ˚ T ˚ Θ p ξ q tT T h T h
MDT ` ´´´`` `` ´ r ` r ` r initial pulse h ´ h looooooooooooomooooooooooooon control D ` MDT D ´ MDT
Figure 3.7:
Terminology for constructing an inverse of I ´ σ ‹ Rσ ‹ R . Here Θ is a halfspace t x n ą u and c ispiecewise constant with discontinuities along planes of constant x n (dashed lines). The wavefront set of the initialpulse h is a single ray; to isolate h MDT three additional singularities are added to h as indicated. Returning, p`q -,and p´q -escapable bicharacteristics are labeled r , ` , and ´ respectively. ii. all of its connecting bicharacteristics at t ˘ are p˘q -escapable;iii. one of its connecting bicharacteristics at t ˘ is p˘q -escapable, and the opposite bicharacteristic is p¯q -escapable.In the final case, if the p˘q -escapable connecting bicharacteristic is a reflection, we also require c to bediscontinuous at lim t Ñ t ˘ γ p t q to ensure the reflection operator has nonzero principal symbol there.Roughly speaking, we may ensure a singularity traveling along a p`q -escapable bicharacteristic nevercreates a singularity in D ` MDT by choosing h appropriately. Similarly, we may produce a singularity along a p´q -escapable bicharacteristic without introducing any extra singularities inside D ` MDT .Now, if every returning bicharacteristic in WF p F h q is p`q -escapable, we can find an h isolating h MDT with an FIO construction, leading to a microlocal inverse of I ´ σ ‹ Rσ ‹ R . Accordingly, let S Ă T ˚ Θ be theset of ξ R W such that every returning bicharacteristic belonging to a broken bicharacteristic through ξ is p`q -escapable . We then have the following result: Proposition 3.2.
There is an FIO A : E p Θ q ‘ E p Θ q Ñ D p R n q ‘ D p R n q of order 0 satisfying p I ´ σ ‹ Rσ ‹ R q A ” I on D S . (3.15) Furthermore, R T Ah ” R T h MDT for any WF p h q Ă S . Note that, because any broken ray intersects only finitely many interfaces in the time interval t P r , T s ,the condition of being p˘q -escapable is open, and in particular S is open. With the microlocal inverse A constructed for I ´ σ ‹ Rσ ‹ R (knowing c ), we may now examine the behaviorof Neumann iteration (which does not require knowing c ). Recalling (3.5), define the Neumann iteration Recall from §3.1 that W is the set of covectors for which the parametrix ˜ R is valid. N k “ k ÿ i “ p σ ˚ ˜ R q i . (3.16)In this section we present our main microlocal theorem: the operators N k isolate h MDT in a particular leadingorder sense as k Ñ 8 . Throughout, as in (3.16) we substitute for R the parametrix ˜ R having cutoffs nearglancing rays.Since lim N k has no microlocal interpretation in general we will instead consider the convergence of thepartial sum operators’ principal symbols. Technically, of course, these symbols belong to separate spaces,since each N k is associated with a different Lagrangian in general. Hence, we first define a suitable symbolspace containing the principal symbols of A and N k , and any reasonable FIO parametrix of (3.4). We thenintroduce a natural (cid:96) norm, which acts as a microlocal energy norm , on restrictions of the symbol space, andstate the convergence theorem.To describe the principal symbols of A and N k , we split them into finite sums of Ψ DOs composed withfixed unitary FIO, then record the Ψ DOs’ principal symbols; this is a kind of polar decomposition. Asis well-known (see appendix A), after a standard microlocal splitting of the wave equation into positiveand negative wave speeds, ˜ R is a sum of graph FIO R s , one for each finite sequence s P t R , T u j , j ě ofreflections and transmissions. For each s , let C s be the canonical transformation of R s ; form the set of allpossible compositions C “ t C s p q ˝ ¨ ¨ ¨ ˝ C s p m q | m ě u . (3.17)and enumerate this resulting set with a single index i : C “ t C i | i P I u . (3.18)Hence, each composition of reflections, transmissions, and time-reversals leads to a canonical transformation C i ; in general, a single C i might be represented by (infinitely many) different compositions C s p q ˝ ¨ ¨ ¨ ˝ C s p m q .We term an FIO C -compatible if it is associated with a finite union of C i .Next, fix a set of elliptic FIO p J i q i P I associated with the C i that are microlocally unitary, that is, J ˚ i J i ” I .Any C -compatible FIO Z may now be written in the form Z “ ř i P I P i J i for appropriate Ψ DOs P i . Definethe principal symbol of Z with respect to p J i q i P I to be the tuple of principal symbols of the P i , restricted to thecosphere bundle: σ “ σ p Z q “ ` σ p P i q ˘ i P I P C ` S ˚ p RRR n z Γ q ˆ I ˘ , (3.19)The boldface RRR n z Γ denotes a doubled space containing two copies of R n z Γ ; due to the microlocal splittingthis is a natural space for Cauchy data. For convenience, we consider the tuple σ as a function on a singledomain having one copy of S ˚ p RRR n z Γ q for each i P I . Note that a full symbol for Z (not needed here) couldbe defined analogously.Now, for η P S ˚ p RRR n z Γ q define G η “ tp C i p η q , i q | i P I , η P D p C i qu Ă S ˚ p RRR n z Γ q ˆ I , (3.20)where D p C i q is the domain of C i . That is, G η contains all covectors reachable from η , together with aknowledge of the paths i taken for each.Consider the restriction of a principal symbol σ p Z q to the space G η . Here, σ p Z q may be viewed both asan element of G η and the unique linear operator on G η defined by left-composition: σ p Z q : σ p Z q ˇˇ G η ÞÑ σ p ZZ q ˇˇ G η , (3.21)for C -compatible FIOs Z . The composition ZZ is well-defined as an FIO since all operators involved aresums of graph FIO.The key idea is that the (cid:96) norm on G η provides a natural microlocal energy operator norm for Z . Inparticular (see Lemma 3.6 in §3.7), just as (cid:107) R (cid:107) “ w.r.t. the exact operator norm, so composition with ˜ r hasoperator norm 1 on the (cid:96) p G η q principal symbol space, in the absence of glancing ray cutoffs. Combining this22orm with existence of an (cid:96) -bounded microlocal inverse of I ´ σ ‹ ˜ Rσ ‹ ˜ R , we can prove principal symbolconvergence for Neumann iteration. In the limit, furthermore, the wave field produced by Neumann iterationat t “ T inside Θ agrees with that produced by the given microlocal inverse, modulo C . Theorem 3.3.
Suppose ˜ S Ă ˚ T ˚ p RRR n z Γ q is a conic set on which I ´ σ ‹ ˜ Rσ ‹ ˜ R has a C -compatible right parametrix ˜ A on ˜ S ; that is, p I ´ σ ‹ ˜ Rσ ‹ ˜ R q ˜ A ” I on ˜ S . Assume that σ p ˜ A q restricts to a bounded operator on (cid:96) p G η q for each η P ˜ S X S ˚ p RRR n z Γ q .Then, for every η P ˜ S X S ˚ p RRR n z Γ q , the Neumann series principal symbols σ p N k q converge to some n P (cid:96) p G η q .Furthermore, σ p ˜ RN k q Ñ σ p ˜ R ˜ A q in (cid:96) p G η X S ˚ Θ q . Of course, we have in mind for ˜ A the concrete parametrix A of Proposition 3.2. This parametrix is C -compatible (cf. §3.7.2); it also has finitely many graph FIO components, so it is a bounded operator on (cid:96) p G η q . Taking ˜ A “ A we have the following direct corollary of Proposition 3.2 and Theorem 3.3: Corollary 3.4.
For every η P S X S ˚ p RRR n z Γ q , the Neumann series principal symbols σ p N k q converge in (cid:96) p G η q .Furthermore, σ p RN k q Ñ σ p RA q in (cid:96) p G η X S ˚ Θ q . According to Proposition 3.2, we have R T Ah ” R T h MDT on T ˚ Θ . Hence, the corollary implies that toleading order, the same is true of the N k as k Ñ 8 ; they also isolate h MDT .Note that Theorem 3.3 does not claim that the principal symbol limit n is itself the principal symbol ofsome FIO. In particular, the support of n on some fiber G η may be infinite, that is, n maps η to infinitelymany singularities. In this case it is not obvious that n corresponds to any FIO. Conversely, if n is smoothand its restriction to every G η has finite support, an FIO N with principal symbol n is easily constructed. The previous two sections treated the solution of p I ´ σ ‹ Rσ ‹ R q h ” h , both constructively and iteratively.In this section we turn to the question of uniqueness; i.e. the solutions of g ” σ ‹ Rσ ‹ Rg . As we will see,the microlocal scattering control equation displays two distinct kinds of nonuniqueness: a normal type,due to diving rays and total reflections, and a pathological type, involving an infinite-energy sequence ofreinforcing singularities.The first type is analogous to the nonuniqueness seen in the exact setting. In the exact case, the kernel G of I ´ π ‹ Rπ ‹ R consists only of initial data whose wave fields are supported outside Θ , due to unique contin-uation. In other words, no waves can enter Θ , completely reflect, and leave in finite time T . Microlocally,however, there is a much richer space of completely reflecting wave fields, including totally reflecting anddiving rays. Note that these rays do not affect h ˇˇ Θ and in particular do not interfere with the wave field of h MDT , up to smoothing.The second type of nonuniqueness is unique to the microlocal setting. In this case, the wave fieldproduced by initial data g does include singularities inside Θ at time T , which σ ‹ cuts off. The (microlocal)energy lost in this cutoff must be replenished by a second singularity in the initial data, which in turn mustbe replenished a third, and so on, necessitating an infinite chain of singularities. Since Rg is not smooth in Θ , the converse of Lemma 3.1 fails.In the following examples, we illustrate these two nonuniqueness types at length. Example 3.1.
Figure 3.8(a) presents an element of the microlocal kernel of p I ´ σ ‹ Rσ ‹ R q , with a diving ortotally reflecting ray and one interface. If g has singularities at a and b satisfying an appropriate pseudodif-ferential relation, its wave field will be smooth along the dashed ray. Thus the cutoffs σ ‹ have no effect, and σ ‹ Rσ ‹ Rg ” RRg “ g , implying p I ´ σ ‹ Rσ ‹ R q g ” .Figure 3.8(b) illustrates how this lack of injectivity leads to multiple solutions h . Here, a stray ray fromthe direct transmission can be cancelled by an appropriate singularity at either a or b , or a linear combinationof them. The proof of Theorem 3.3 shows that Neumann iteration converges in principal symbol to a solutionoperator having “least microlocal energy” in the sense of a weighted (cid:96) norm on its principal symbol.23 ∗ T ∗ Θ ( ξ ) ab ∂T ∗ Θ (a) An element in the microlocal kernel of p I ´ σ ‹ Rσ ‹ R q t T T ab h MDT h d ∗ T ∗ Θ ( ξ ) D + MDT ∂T ∗ Θ (b) Nonuniqueness for microlocal scattering control
Figure 3.8:
Regular nonuniqueness for microlocal scattering control; interfaces are marked with discs. (a) Anappropriate combination of singularities at a and b is smooth on the dashed bicharacteristic and reflects from Θ . (b)A singularity from h can be cancelled at either a or b . t T a a a b b x . . . Figure 3.9:
One-dimensional example of pathological nonuniqueness. Θ is a union of infinitely many intervals;dotted lines are interfaces. The pattern continues indefinitely as x Ñ `8 . Θ a a a · · · b b · · · Figure 3.10:
Two-dimensional version of Figure 3.9. Thin lines represent interfaces; dashed rays never reach thesurface. Total internal reflection occurs at the upper interface. xample 3.2. Figure 3.9 shows a one-dimensional setup exhibiting the second type of nonuniqueness. Whilethis example is contrived, Figure 3.10 shows how an equivalent and more realistic higher-dimensional versionmay be constructed. (Both examples involve non-compact domains, and we conjecture noncompactness isrequired for this type of nonuniqueness.)Here Θ consists of an infinite series of disconnected open intervals p´8 , w q Y p v , w q Y p v , w q Y ¨ ¨ ¨ .On each finite interval c has two jump discontinuities; assume Θ is sufficiently close to Θ to contain thesesingularities. Two sequences of unit covectors t a i u i “ , t b i u i “ Ă S ˚ Θ ‹ z W are chosen so that the canonicalrelation of σ ‹ ˜ R sends a i to t b i , b i ` u and b i to t a i ´ , a i u .We now construct a g in the microlocal kernel of I ´ σ ‹ Rσ ‹ R with an infinite sequence of singularities at a , a , a , . . . . First, note that the canonical relation of σ ‹ ˜ Rσ ‹ ˜ R sends a i p i ą q to t a i ´ , a i , a i ` u . Supposenow that we choose some initial data g with a singularity at a . After applying σ ‹ Rσ ‹ R , some portion of thissingularity’s amplitude will be lost due to the σ ‹ cutoffs. We may, however, restore the lost amplitude byadding an appropriate singularity to g at a . In turn, some of this new singularity’s amplitude will be lostunder σ ‹ Rσ ‹ R , which we make up for with an appropriate singularity at a , and so on.Rigorously, decompose σ ‹ ˜ Rσ ‹ ˜ R near each a i as the sum of three graph FIO A ´ , A , A whose canonicalgraphs map a i to a i ´ , a i , and a i ` respectively. Modify A , say, by a smooth operator so that σ ‹ Rσ ‹ R “ A ´ ` A ` A exactly. It can be shown (cf. (A.4)) that the A k are elliptic.Now, choosing any g P L p Θ ‹ q with WF p g q “ R ` a , we look for g i , i “ , , . . . with wavefront sets at R ` a i such that the sum g “ ř g i satisfies p I ´ σ ‹ Rσ ‹ R q g ” . This leads to the infinite matrix equation ¨˚˚˚˚˚˝ I ´ »—————– A A ´ A A A ´ A A . . .. . . . . . fiffiffiffiffiffifl˛‹‹‹‹‹‚»—————– g g g ... fiffiffiffiffiffifl ” . (3.22)By ellipticity, (3.22) has a solution, namely g i ` ” p A ´ q ´ ` p I ´ A q g i ` A g i ´ ˘ . To construct anassociated g , we use the fact that the t a i u are discrete in S ˚ p Θ ‹ q (which implies Θ is unbounded).Each g i is locally L , so after multiplying by a smooth cutoff near the base point of a i , we may assume g i P L . Applying radial cutoffs in the Fourier domain, we may assume that (cid:107) g i (cid:107) L ď ´ i , so g “ ř g i converges in L . Defining g ´ “ , consider p I ´ σ ‹ Rσ ‹ R q g “ ÿ i “ ´ A g i ´ ` p I ´ A q g i ´ A ´ g i ` . (3.23)Each summand is smooth by construction, and compactly supported near the base point of a i . Becausethe t a i u are discrete, we can ensure only finitely many summands of (3.23) are nonzero at any given point.Hence the entire sum is smooth, showing g is in the microlocal kernel of I ´ σ ‹ Rσ ‹ R . As expected, Rg is notsmooth in Θ ; it is not hard to see it must be singular at every b i . Hence, solving p I ´ σ ‹ Rσ ‹ R q h ” h is notsufficient for isolating h MDT . Uniqueness and Isolating h MDT
We now close the circle, and return to the question of whether solving p I ´ σ ‹ Rσ ‹ R q h ” h is equivalent to isolating h MDT . Of our two types of nonuniqueness, only the secondinterferes with isolating h MDT . We may rule it out, to leading order, by assuming the same kind of microlocalenergy boundedness seen earlier in Theorem 3.3: namely, (cid:96) boundedness of the parametrix’s principalsymbol. Assuming this condition, we reach a partial converse of Lemma 3.1: a solution of the microlocalscattering control equation isolates h MDT to leading order as long as this is possible. We frame our propositionas a uniqueness result.
Proposition 3.5.
Suppose B , B are C -compatible microlocal right inverses for I ´ σ ‹ ˜ Rσ ‹ ˜ R on a conic subset ˜ S Ă ˚ T ˚ p RRR n z Γ q . If their principal symbols restrict to elements of (cid:96) p G η q for all η P ˜ S , ˜ RB h ˇˇ Θ ” ˜ RB h ˇˇ Θ mod H s ` p RRR n z Γ q for all h P H s p RRR n z Γ q X D ˜ S . (3.24)25n particular, as long as there is some “finite microlocal energy” parametrix isolating h MDT on a conic set ˜ S Ă ˚ T ˚ p RRR n z Γ q , all other finite microlocal energy parametrices on ˜ S also isolate h MDT . The major task in proving Theorem 3.3 is to show that composition with ˜ R has operator norm at most 1 on (cid:96) p G η q for any η — a microlocal version of energy conservation. We begin with its proof.To present the energy conservation lemma, note that composition with ˜ R is linear and well-definedon C -compatible FIO. It therefore induces a linear operator ˜ r on their principal symbols in the space C p S ˚ p RRR n z Γ q ˆ I q . Since G η is closed under the canonical relation of ˜ R , operator ˜ r restricts to a linearoperator on (cid:96) p G η q for any η P S ˚ p RRR n z Γ q . Lemma 3.6 (Microlocal Energy Conservation) . Let η P S ˚ p RRR n z Γ q . Then (cid:107) ˜ r (cid:107) ď with respect to the operatornorm on (cid:96) p G η q .Proof. First, assume that there are no cutoffs in the parametrix ˜ R due to glancing rays originating in G η . Inthis case, ˜ R ” R “ I , so ˜ r “ I likewise. If ˜ r were self-adjoint, it would follow that (cid:107) ˜ r (cid:107) (cid:96) “ . Certainly ˜ R is microlocally self-adjoint, since ˜ R ˚ ” R ˚ “ R ” ˜ R . This property does not immediately carry over to ˜ r dueto the presence of Maslov factors; fortunately, it is still possible to show ˜ r is self-adjoint.Let p α, i q , p β, j q P G η , and let e α,i , e β,j P (cid:96) p G η q be the vectors having 1 in the p α, i q or p β, j q positionrespectively and zeros elsewhere. It suffices to show that x ˜ re α,i , e β,j y “ x ˜ re β,j , e α,i y . (3.25)To compute each side, we choose Ψ DOs
P, P P Ψ with σ p P q “ σ p P q “ near α, β respectively.Decompose ˜ RP J i ” ÿ j P I Q j J j , ˜ RP J j ” ÿ i P I Q i J i . (3.26)The left- and right-hand sides of (3.25) then become σ p Q j qp β q and σ p Q i qp α q .If there is no C s carrying p α, i q to p β, j q (that is, C s p α q “ β and C s ˝ C i “ C j on their common domainof definition), there is also no C s carrying p β, j q to p α, i q , and vice versa. In this case, both sides of (3.25)are zero. Otherwise, there are unique C s and C s satisfying the above; let R s and R s be the microlocalrestrictions of ˜ R to each of these canonical relations near α and β respectively. We may replace ˜ R in the firstand second equations of (3.26) by R s and R s , respectively. Furthermore, R s ” R ˚ s since ˜ R is microlocallyself-adjoint and C s “ p C s q ´ .Now we apply singular symbol calculus (see [5]) to both sides of the first equation of (3.26) and evaluateat β and α . Let lowercase letters ( r s , j i , etc.) denote singular principal symbols (of R s , J i , etc.). This yields r s p β q j i p η q i κ p d C i p V η q , V α , dC ´ s p V β qq{ “ q j p β q j j p η q ,r s p α q j j p η q i κ p d C j p V η q , V β , dC s p V α qq{ “ q i p α q j i p η q , (3.27)where V γ denotes the vertical subspace in T γ T ˚ p RRR n z Γ q , and κ is the Kashiwara index [12, 15]. Solving for q j p β q and q i p α q we obtain x ˜ re α,i , e β,j y “ q j p β q “ r s p β q j i p η q j j p η q i ´ κ p d C i p V η q , V α , dC ´ s p V β qq{ , x ˜ re β,j , e α,i y “ q i p α q “ r s p α q j j p η q j i p η q i κ p d C j p V η q , V β , dC s p V α qq{ . (3.28)26omparing terms, r s p β q “ r s p α q since R s “ R ˚ s , and similarly j i p η q{ j j p η q “ j j p η q{ j i p η q , because J i beingunitary implies | j i | “ . As for the Kashiwara indices, since κ is coordinate-invariant and alternating, κ p d C i p V η q , V α , dC ´ s p V β qq “ κ p d C j p V η q , dC s p V α q , V β q“ ´ κ p d C j p V η q , V β , dC s p V α qq . (3.29)The conclusion is that ˜ r is self-adjoint, and therefore (cid:107) ˜ r (cid:107) “ , since (cid:107) ˜ r (cid:107) “ (cid:107) I (cid:107) “ .In the presence of near-glancing rays in G η , the parametrix constructed in appendix A includes pseu-dodifferential cutoffs away from glancing rays (in constructing ϕ ` and J B (cid:1) S ). In a neighborhood of any α P G η for which some broken ray is at least partially cut off, ˜ R is microlocally equivalent to a composition ofpropagators and pseudodifferential cutoffs ˜ R ” υ ˝ ˜ R t m ˝ P m ´ ˝ ˜ R t m ´ ˝ ¨ ¨ ¨ ˝ P ˝ ˜ R t , (3.30)where t ` ¨ ¨ ¨ ` t m “ T and P , . . . , P m ´ P Ψ have principal symbols of magnitude at most 1, and noneof the intermediate propagators ˜ R t k involve glancing ray cut offs when ˜ R is restricted to the neighborhoodof α .For each k “ , . . . , m , we let C p k q “ t C p k q s ˝ C i u be the set of compositions of C i ’s with canonical graphs C p k q s defined as in §3.5 but with T replaced by t ` ¨ ¨ ¨ ` t k . Naturally, C p q “ C p m q “ C . Choose sets ofcorresponding unitary operators t J p k q i u as before for each k . Then composition by each ˜ R t k sends C p k q - to C p k ` q -compatible FIO, and as before induces a map between their principal symbol spaces; the argumentabove shows it is an isometry with respect to the (cid:96) norms.Composition with the pseudodifferential cutoffs P k acts by pointwise multiplication by p k on these (cid:96) spaces, and hence has operator norm at most 1. Since C p m q “ C , operator ˜ r is given by the composition ofall these operators ˜ r t m ˝ p m ´ ˝ ˜ r t m ´ ˝ ¨ ¨ ¨ , and thus (cid:107) ˜ r (cid:107) ď . Proof of Theorem 3.3.
We begin with the first statement of the theorem: convergence of the N k ’s principalsymbols in (cid:96) p G η q .Since composition with σ ‹ multiplies principal symbols pointwise by σ ‹ , it is a linear operator on (cid:96) p G η q with norm at most 1. Therefore σ ‹ ˜ rσ ‹ ˜ r , the operation of principal symbol composition with σ ‹ ˜ Rσ ‹ ˜ R , hasnorm at most 1 as an operator on (cid:96) p G η q .Let n k , ˜ a , and i denote the principal symbols of N k , ˜ A , and the identity with respect to the J i . We will seethat ˜ a ’s existence implies the convergence of n k by the spectral theorem, applied to a symmetrization of σ ‹ ˜ r .Restricting to G η , suppose p I ´ σ ‹ ˜ rσ ‹ ˜ r q u “ i for some u P (cid:96) p G η q . (3.31)Then u “ i ` v for some v in the range of σ ‹ . In particular, v is supported in G η X T ˚ Θ . Solving (3.31) for w “ v {? σ ‹ gives p I ´ ? σ ‹ ˜ rσ ‹ ˜ r ? σ ‹ q v ? σ ‹ “ ? σ ‹ ˜ rσ ‹ ˜ ri. (3.32)As the process is reversible, u is a solution of (3.31) if and only if w “ p u ´ i q{? σ ‹ solves (3.32) in the weightedspace (cid:96) p G η X T ˚ Θ , σ ‹ q . Now, if there is any solution to (3.32), applying Lemma 2.11 to the self-adjointoperator ? σ ‹ ˜ r ? σ ‹ shows that the Neumann series w “ ÿ k “ “ ? σ ‹ ˜ rσ ‹ ˜ r ? σ ‹ ‰ k ? σ ‹ ˜ rσ ‹ ˜ ri (3.33)converges in (cid:96) p G η X T ˚ Θ , σ ‹ q to the minimal-norm solution of (3.32). The corresponding u “ i ` ? σ ‹ w P (cid:96) p G η q is exactly lim n k .In particular, u “ ˜ a is a solution of (3.31) and it is in (cid:96) p G η q since its support in G η is finite. Hence, theNeumann series partial sum principal symbols converge in (cid:96) p G η q . They may not converge to ˜ a , as I ´ σ ‹ ˜ rσ ‹ ˜ r may have a nontrivial nullspace. 27onsider this nullspace. Suppose p I ´ σ ‹ ˜ rσ ‹ ˜ r q g “ for some g P (cid:96) p G η q , so that g “ σ ‹ ˜ rσ ‹ ˜ rg . But sincethe operator norms of σ ‹ and ˜ r are at most 1, we must have (cid:107) g (cid:107) “ (cid:107) ˜ rg (cid:107) “ (cid:107) σ ‹ ˜ rg (cid:107) “ (cid:107) ˜ rσ ‹ ˜ rg (cid:107) “ (cid:107) σ ‹ ˜ rσ ‹ ˜ rg (cid:107) . (3.34)The second equality implies that ˜ rg is supported in T ˚ Θ . Taking g “ ˜ a ´ lim n k , we conclude ˜ ra and ˜ r ˝ lim n k are equivalent in T ˚ Θ , finishing the proof. Proof of Proposition 3.2.
The proof is purely technical, specifying a recursive procedure for constructing aset of incoming singularities that ensure that only the directly-transmitted singularity reaches D ` MDT . Thenotation of Appendix A will be used throughout.Our key constructions will be order-0 FIO Ξ i ˘ , Ξ o ˘ : C p R ˆ B Z q Ñ D p Z q producing tails outside Θ for p˘q -escapable bicharacteristics. Following §3.4, the Ξ i { o ` -constructed tail for a singularity on a p`q -escapablebicharacteristic ensures this singularity escapes Θ at time T , without generating any singularities in h MDT ’smicrolocal forward domain of influence, D ` MDT . The Ξ i { o ´ -constructed tail generates a given singularity on a p´q -escapable bicharacteristic, again without causing any singularities to enter D ` MDT . The Ξ o ˘ are defined onoutgoing boundary data while the Ξ i ˘ are defined on incoming data, microlocally near the final, resp., initialcovectors of p˘q -escapable bicharacteristics.Let γ : p t ´ , t ` q Ñ T ˚ Z be a p˘q -escapable bicharacteristic. Denote by β o the pullback to the boundary ofits final point: β o “ p di Γ q ˚ γ p t ˘ q , where by abuse of notation we consider γ p t ˘ q as a space-time covector, in ˚ T ˚ p R ˆ Z q . Define β i “ p di Γ q ˚ γ p t ¯ q similarly. We now define Ξ i { o ˘ microlocally near β i { o , starting with theincoming maps Ξ i ˘ .• If t ˘ P p , T q : We simply follow the bicharacteristic and apply Ξ o ˘ at the other end. In the p`q casedefine Ξ i ` ” Ξ o ` J B (cid:1) B near β i . In the p´q case, define Ξ ´ ” Ξ ´ J ´B (cid:1) B M near β i , where J ´B (cid:1) B “ υJ B (cid:1) B υ islike J B (cid:1) B but propagating backward in time.• If γ escapes, t ˘ R r , T s : This is the terminal case. In the p`q case, there is nothing to do: define Ξ ` ” near β i . For the p´q case, define Ξ ´ ” J ´ C (cid:1) B near β i to obtain the necessary Cauchy data.We now turn to Ξ o ˘ , considering each case in the definition of p˘q -escapability.• If γ escapes: This case never arises: Ξ i ˘ is not defined in terms of Ξ o ˘ for such γ .• If all outgoing bicharacteristics are p˘q -escapable:
Recursively apply Ξ i ˘ to the reflected and transmitted (ifany) bicharacteristics, defining Ξ o ˘ ” Ξ i ˘ M near β o .• If one outgoing bicharacteristic is p˘q -escapable, and the opposite incoming ray is p¯q -escapable:
This is thecore case. In the p`q case, near β o let Ξ o ` ” ´ Ξ i ´ M ´ R M T ` Ξ i ` p M R ´ M T M ´ R M T q , case (R), ´ Ξ i ´ M ´ T M R ` Ξ i ` p M T ´ M R M ´ T M R q , case (T), (3.35)according to whether the reflected (R) or transmitted (T) outgoing ray is p`q -escapable. The inversesare all microlocal. The p´q case is slightly different: near β o , Ξ o ´ ” Ξ i ´ M ´ R ` Ξ i ` M T M ´ R , case (R), Ξ i ´ M ´ T ` Ξ i ` M R M ´ T , case (T). (3.36)For case (R), the requirement in the definition that c be discontinuous at β i { o implies that M R ’s principalsymbol is nonzero there (cf. (A.4)), guaranteeing the existence of a parametrix M ´ R near β i { o . For case(T), M T always has positive principal symbol, regardless of c .28hile Ξ i { o ˘ is defined recursively, by definition only finitely many recursions are needed to reach thenon-recursive case where γ escapes. Since all the cases are open conditions on β , operators Ξ i { o ˘ are well-defined (assuming that in regions where both the second and third cases hold, we decide between themconsistently). Furthermore, the Ξ i { o ˘ are order-0 FIO, since they are microlocally sums of compositions oforder-0 FIO associated with invertible canonical graphs.We now use Ξ i { o ˘ to define a parametrix A . Given η P S Ă ˚ T ˚ Θ , consider the escaping bicharacteristicsstarting at η . Each is associated with a distinct sequence of reflections and transmissions s “ p s , . . . , s k q Pt R, T u k for some k , and a corresponding propagation operator P s “ J B (cid:1) B M s k ¨ ¨ ¨ J B (cid:1) B M s J B (cid:1) B M s J C (cid:1) B . (3.37)Let S be the set of escaping bicharacteristic sequences s , and define A η “ I ` Ξ o ` ÿ s P S P s , (3.38)Then define A by patching together the A η with a microlocal partition of unity. As Ξ i { o ˘ , P s are FIO of order 0,so is A .We now check that A isolates h MDT and is therefore a microlocal right inverse for I ´ σ ‹ Rσ ‹ R byLemma 3.1. Let h be microsupported in a sufficiently small neighborhood of η P S and let h “ Ah . Definethe outgoing boundary parametrix B “ J B (cid:1) S ÿ k “ p M J B (cid:1) B q k . (3.39)With P s , S as before, define S K to be the set of sequences s for which no s P S is a prefix. Then ˜ F h splitsinto three components: ˜ F h “ ˜ F p h ´ h q ` B M ÿ s P S P s h ` ÿ s P S K ˜ F s . (3.40)For t P r T, T s , the last term is the wave field of h MDT ; accordingly, it suffices to prove that the sum of firsttwo terms are smooth in D ` MDT . Rewrite ˜ F p h ´ h q ` B M ÿ s P S P s h “ ÿ s P S p ˜ F Ξ o ` ` B M q P s h . (3.41)By construction, ˜ F Ξ o ` ` B M is smoothing at the terminal end of p`q -escapable bicharacteristics, and inparticular on WF p P s h q for each s P S , as desired. Hence ˜ R T h ” ˜ R T h MDT . Applying Lemma 3.1, weconclude p I ´ σ ‹ ˜ Rσ ‹ ˜ R q Ah ” h . The same result holds for all h P D S by a microlocal partition of unity. Proof of Proposition 3.5.
Let b , b , i be the principal symbols of B , B , and the identity. Letting σ ‹ and ˜ r denote the operators on the space of principal symbols induced by multiplication with σ ‹ and compositionwith ˜ R , respectively, p I ´ σ ‹ ˜ rσ ‹ ˜ r qp b ´ b q “ . As in the proof of Theorem 3.3, it follows that ˜ r p b ´ b q issupported in T ˚ Θ . Both the exact analysis of Section 2 and the microlocal analysis of Section 3 prove that scattering controlisolates a certain portion of the wave field of h at t “ T , while effectively erasing the rest. Our two analyses,however, predict the isolation of two different portions of the wave field. Surprising at first glance, thisdisparity provides further insight on scattering control, which we explore in this section.While the arguments are quite general, we consider for simplicity two particular examples that illustratethe fundamental differences between dimensions n “ and n ą . In the one-dimensional example,29 Θ ∂ Θ T η> T < T Figure 4.1:
A singularity in h MDT but not h DT . Its distance along the slanted bicharacteristic is greater than T , but itsbase point is less than distance T from the boundary. Hence η P p T ˚ Θ q T but η R T ˚ p Θ T q . the microlocal and exact analyses align as h DT and h MDT are essentially equal; the result is unconditionalconvergence of the Neumann iteration, both exactly and microlocally. In higher dimensions, however, h DT and h MDT can be quite different, causing a loss of convergence in finite energy space. n “ dimension Let Ω “ p (cid:15), and Θ “ p , for fixed (cid:15) ą ; let Θ , Θ be arbitrary. Let c be piecewise smooth on R , andequal to 1 on Ω ‹ . In general, the distance of a point from B Θ is the minimum distance of a singularity at thatpoint from B Θ : d p x, B Θ q “ min ξ P ˚ T ˚ x R d p ξ, B T ˚ Θ q . (4.1)In one dimension, this means d ˚ T ˚ Θ p ξ q “ d ˚ Θ p x q if ξ P ˚ T ˚ x R . Hence, h DT and h MDT are essentially equivalent,differing only in their respective usage of harmonic extensions and smooth cutoffs. We now discuss themicrolocal and exact behaviors that arise in scattering control.On the microlocal side, (4.1) implies every returning bicharacteristic is trivially p`q -escapable, as noglancing or totally reflected waves arise. Consequently, the constructive parametrix A may be definedeverywhere in ˚ T ˚ Θ , and hence by Theorem 3.3 microlocal Neumann iteration always converges in principalsymbol.On the exact side, the exact Neumann series converges to a finite energy solution h of p I ´ π ‹ Rπ ‹ R q h “ h , thanks again to microlocal analysis. To see why, first separate the initial data into rightward- and leftward-traveling waves (possible since c “ there). The rightward-traveling portion has a directly transmittedcomponent inside Θ , which is its image under an elliptic graph FIO. Due to the ellipticity this directlytransmitted wave carries a positive fraction of the initial energy, by Gårding’s inequality and uniquecontinuation (compare Stefanov and Uhlmann’s work [17]). Leftward-traveling waves, meanwhile, may besafely ignored, since c is constant for x ă . The full proof requires some care, and we defer it to §4.3. Proposition 4.1.
Let Ω , Θ , c be as above, and (cid:15) ă T . Then (cid:107) π ‹ Rπ ‹ R (cid:107) ă on H p Ω ‹ q ‘ L p Ω ‹ q ; in particular ř k “ p π ‹ R q k h always converges. n ą dimensions Consider a halfspace Θ “ t x n ě u , and let c p x q “ . Any η “ p x , x n , ξ , ξ n q P ˚ T ˚ Θ with x n ą T thenbelongs to T ˚ p Θ T q . However, if ξ ‰ , then d ˚ T ˚ Θ p η q ą x n and η R p T ˚ Θ q T if T is sufficiently close to x n (Figure 4.1). This discrepancy, which of course occurs for general Θ , c when n ą , implies that h DT isfundamentally smaller than h MDT . Furthermore, it prevents the exact Neumann series from converging (infinite energy space) for any h producing singularities in the gap p T ˚ Θ q T z T ˚ p Θ T q , as we now show.Suppose η P WF p R T h q X ` p T ˚ Θ q T z T ˚ p Θ T q ˘ , and γ is the bicharacteristic passing through η at t “ T . Ifthere were a finite energy solution h P C of the scattering control equation (2.12), the proof of Theorem 2.2implies (via unique continuation) that the wave field v p t, x q “ p F πRh qp T ´ t, x q is stationary harmonicat t “ T on Θ ‹ T , and in particular smooth at η . Propagation of singularities makes this impossible, since γ pr , T sq lies completely inside Θ . Hence no h P C exists, and the Neumann series for h must diverge,implying that (cid:107) π ‹ Rπ ‹ R (cid:107) “ . 30sing this argument, a divergent Neumann series may be constructed whenever p T ˚ Θ q T ‰ T ˚ p Θ T q .Hence we expect (cid:107) π ‹ Rπ ‹ R (cid:107) “ in general for n ą dimensions, in opposition to Proposition 4.1 in 1D. Itis worth noting that in numerical tests the Neumann iteration appears to follow its microlocally predictedbehavior (isolation of h MDT ) more closely than its exact behavior (isolation of h DT ). Proof of Proposition 4.1.
This proof is inspired in large part by a proof of Stefanov and Uhlmann [17, Prop.5.1]. Let x p t q be the inverse function of the travel time t “ ş x c p x q ´ dx “ d ˚ Θ p x q ; then Θ t “ p x p t q , .Choose δ ą small enough that | t ´ t | ą δ { for any distinct x p t q , x p t q P singsupp c .In p´8 , (cid:15) q take the factorization B t ´ ∆ “ pB t ` i B x qpB t ´ i B x q associated with d’Alembert solutions u p t, x q “ f p x ´ t q ` g p x ` t q . Identifying h with p f, g q P H ˆ H , (cid:107) h (cid:107) “ ż (cid:15) c ´ (cid:12)(cid:12) g ´ f (cid:12)(cid:12) ` (cid:12)(cid:12) f ` g (cid:12)(cid:12) dx “ ` (cid:13)(cid:13) f (cid:13)(cid:13) L ` (cid:13)(cid:13) g (cid:13)(cid:13) L ˘ . (4.2)The leftward-traveling component g is trivially handled, since it is preserved by Rπ ‹ R : indeed, if f “ ,then supp Rh Ă p´ T, ´ T ` (cid:15) q , and π ‹ Rπ ‹ Rh “ π ‹ R h “ . Hence we restrict attention to rightward-traveling initial data h “ p f, q .Intuitively, the energy of the direct transmission of f , that is, its image under the graph FIO componentsof R involving only transmissions, should be bounded away from zero by Gårding’s inequality since thesecomponents are elliptic.To start, assume supp h is contained in an interval p a, b q of width b ´ a ď δ , so that no multiply-reflectedrays enter the direct transmission region I “ p x p a ` T q , x p b ` T qq . Furthermore, assume c is constant on I ,so that Rh again divides into leftward- and rightward-travelling components F, G .On I we have Rh ” p R ` DT ` R ´ DT q h , where R ˘ DT are elliptic graph FIO (one for each family of bicharac-teristics) associated with propagation along purely transmitted broken bicharacteristics; see Appendix A. Let π ˘ “ p I ˘ iH q be the projections onto positive and negative frequencies (where H is the Hilbert transform),and define the elliptic FIO R DT “ R ` DT π ` ` R ´ DT π ´ . Now on I we have F ” ψ B x R DT B ´ x f . ApplyingGårding’s inequality to the normal operator of B x R DT B ´ x , with an appropriate spatial cutoff, (cid:107) h (cid:107) “ ? (cid:13)(cid:13) f (cid:13)(cid:13) L ď C ? (cid:13)(cid:13) F (cid:13)(cid:13) L p I q ` (cid:13)(cid:13) Kf (cid:13)(cid:13) L “ C p E I p Rh qq { ` (cid:107) ˜ Kh (cid:107) ď C (cid:107) πRh (cid:107) ` (cid:107) ˜ Kh (cid:107) , (4.3)where K, ˜ K are compact operators. In fact, h “ p f, q K ker πR , so the compact error term (cid:107) ˜ Kh (cid:107) may beeliminated. To see this, by unique continuation h “ p f , g q P ker πR implies F h “ along R ˆ B Ω and r (cid:15), T s ˆ B Θ . Since F h “ f p x ´ t q ` g p x ` t q outside Ω , we conclude f “ . Conversely, π p , g q “ sothat ker πR “ tp , g qu K h .Hence on the subspace g “ , for some constant C ą , (cid:107) π ‹ Rπ ‹ R (cid:107) ď (cid:107) π ‹ R (cid:107) ď ´ C . (4.4)and as π ‹ Rπ ‹ R p f, g q “ π ‹ Rπ ‹ R p f, q this proves the result for all h .The same is true even if c is not constant on I , since without affecting π ‹ Rπ ‹ R we may modify c so as tobe constant on some deeper interval p x p T q , , T ą T ` (cid:15) { , and deduce an estimate analogous to (4.3),but at the later time t “ T . By finite speed of propagation and conservation of energy, we can move theestimate back to t “ T to establish (4.3).Finally, if (cid:15) ą δ , it is possible that the direct transmission of a shallower part of h may be cancelled bythat of a deeper part of h , derailing the Gårding estimate. However, if this occurs the shallower and deeperparts of h must be related by an elliptic FIO; therefore, the shallower part’s energy is controlled by thedeeper part’s direct transmission. 31o make a simpler version of this idea rigorous, cover p´ T, (cid:15) q with intervals of width δ : I j “ pp j ´ q δ, jδ q , j “ t ´ T { δ u , . . . , r (cid:15) { δ s “ k. (4.5)Choose f j P H loc with f j “ I j f , where I j denotes the characteristic function. For each j , we have anestimate of the form (4.3) with h “ p , f j q . Let E j “ ? (cid:107) f j (cid:107) L be the energy of f j . Now, let j be thesmallest j for which E j ě C ´ ř i ą j E i ; this is true of j “ k so such a j always exists. By finite speed ofpropagation, the energy of Rh in I “ p x p T ` p j ´ q δ q , x p T ` j δ qq depends only on f i with i ě j .But the direct transmission of f j contributes at least energy ř i ą j E i , so by conservation of energy andGårding’s inequality (cid:13)(cid:13) f j (cid:13)(cid:13) L À E I p Rh q ` (cid:107) ˜ Kh (cid:107) . (4.6)However, we may bound all of f in terms of f j . For, if j ą j certainly (cid:107) f j (cid:107) À (cid:107) f j (cid:107) ; for j ă j , this is alsotrue as E j ğ C ´ E j . Hence (cid:13)(cid:13) f (cid:13)(cid:13) L ă C E I p Rh q ` (cid:107) ˜ Kh (cid:107) , (4.7)with a constant C “ C p C , (cid:15), δ, T q . The remainder of the proof follows as before. In this section, we illustrate the connection between Marchenko’s integral equation and scattering control byfirst generalizing Rose’s focusing algorithm [14] to higher dimensions. This will show how one can eliminatemultiple scattering in higher dimensions to eventually obtain a focused wave. We will start by summarizingRose’s approach in one space dimension to eliminate multiple scattering and obtain a focused wave. Wewill then explain the drawbacks to his approach, and provide our results that generalize his one-sidedautofocusing results to higher dimensions. In addition, the one dimensional case will provide an accurateillustration of the microlocal solution A constructed in Proposition 3.2. This will provide a clear distinctionbetween the scattering control process and Rose’s focusing algorithm where the advantages of scatteringcontrol are readily apparent. Lastly, we will connect our results with the 1D Marchenko equation used tosolve the inverse scattering problem. In [14], Rose tries to focus an acoustic wave (working in R t ˆ R x ) inside a medium occupying t x ą u . Onthe left side, t x ă u , the wave speed is known, say for simplicity. Inside x ă , the total wave field u maydirectly be decomposed into its incoming and outgoing components: u p x, t q “ u in p x, t q ` u out p x, t q . One is given the reflection response operator that we denote R p t q which relates the incoming and outgoingwaves at the boundary t x “ u . By linearity, one has exactly u out p x “ , t q “ ż R p t ´ t q u in p , t q dt . The goal of Rose is to determine a boundary control u in p x “ , t q such that the total wave field u willbe a distribution with support equal to t x “ x f u at time t “ for some focusing point x f ą one isinterested in. Letting t f denote the focusing time, i.e. t f “ d c p , x f q , Rose uses the ansatz u in p x “ , t q “ δ p t ` t f q ` Ω tail p t ; t f q , and then finds an equation that Ω tail must solve in order to obtain focusing.Rose shows that Ω tail must solve (see [14, Equation (8)]) Ω tail p´ t ; t f q ` R p Ω tail p´ t ; t f qq “ ´ R p δ p´ t ` t f qq for t ă t f , (5.1)32here the action of R applied to a test function φ is R φ “ ż R p t ` t q φ p t q dt . (5.2)Equation (5.1) for Ω tail p´ t ; t f q is the Marchenko equation encountered in 1D potential scattering, whichwe will describe in more detail later. Also, if one denotes r “ δ p t ´ t f q and ˜ K tail “ Ω tail p´ t ; t f q , then thisequation reads ˜ K tail ` R ˜ K tail “ ´ R r for t ă t f , Note that this approach relies heavily on the directional decomposition of a wave field into incoming andoutgoing waves. In higher dimensions, such a decomposition may only be done microlocally, and as such,the reflection response operator R Rose would only be defined microlocally (see [19] for a detailed account ondoing this direction decomposition). The seismic literature has avoided this issue by ignoring the presenceof evanescent and glancing waves, so a rigorous mathematical proof to obtain exact focusing in the presenceof conormal singularities in higher dimensions has never been done. The whole point of using Cauchy datarather than boundary data is to avoid such microlocal considerations and obtain an iteration method in anexact sense.Thus, based on the above equations, if we wanted to generalize this to higher dimensions in an exactsense using our Cauchy data setup, one may naively guess that the appropriate equation should be K tail ` π ‹ RK tail “ ´ π ‹ Rr for r , K tail P C , with r having support in Θ and K tail having support outside Θ . Notice that no directionalwave decomposition is necessary to write down this equation. This in fact turns out to be the correct equation,and we provide a rigorous analysis in the next section.
We prove here a generalization to arbitrary dimension of Rose’s equation (5.1) that allows one to eliminatemultiple scattering of the pressure wave field. This is the key step that will allow one to focus a pressurefield or velocity field at a given time. However, to avoid difficult microlocal issues with directional wavedecompositions, we prove a theorem using Cauchy data rather than boundary data. Afterwards, we relatehow this connects to Rose’s algorithm for focusing discussed in the previous section as well as the classicalMarchenko equation, which use boundary control rather than Cauchy data.We now state the following general theorem about eliminating multiple scattering above a certain depthlevel T (given in travel time coordinates) inside the medium, i.e. within Θ ‹ T . Theorem 5.1.
Let u be the solution to the wave equation with Cauchy data r “ r ` K tail P C , where r has supportin Θ , and K tail has support outside Θ . Let T ą .(i) (Necessity) If u p T q has support in Θ T , then necessarily K tail satisfies the following equation K tail ` π ‹ RK tail “ ´ π ‹ Rr (5.3) (ii) (Partial converse) Suppose K tail satisfies K tail ` π ‹ RK tail “ ´ π ‹ Rr . Then Π ‹ T u p T q “ and u p T q| Θ T “ R T r | Θ T . (iii) (Uniqueness of the tail) Any two tails may only differ by Cauchy data that is totally internally reflected, anddoes not penetrate Θ in time T . That is, if K tail ` π ‹ RK tail “ , then K tail “ in C . iv) (Almost Solvability) The set of r P H for which one has a convergent Neumann series solution for K tail , Q : “ t r P H : p I ` π ‹ R q ´ r P C u is dense in H . (Note that Π ‹ T denotes the orthogonal projection from H p Θ ˚ T q onto H p Θ ˚ T q .) Remark.
The main content of this theorem is that once r is given, then one has a formula to construct K tail that controls the multiple scattering inside Θ ‹ T at time T . The construction of K tail gives no information onwhat happens inside Θ T at time T since K tail does not affect this region. What happens inside Θ T is entirelydetermined by r . Thus, for the purposes of focusing, one needs to construct r beforehand such that theassociated pressure field restricted to Θ T at time T will have a singular support at a single point. In Wapenaaret al. [24], the authors assume they have an approximate velocity profile to construct an approximation to thedirect transmission (denoted T inv d in equation (16) there), which is analogous to the r we have here. Theythen construct a tail (denoted by M ) analogous to our K tail to control the multiple scattering. Remark.
Notice that this theorem never mentions a focusing point but rather an inside region Θ T . Thisis because in order to make the theorem more general, we did not specify any support conditions for r .Typically however, one sends an incident pulse r that is supported close to but outside Ω , which is meant tobe the direct transmission. Then the domain of influence of r inside Θ T at time T is only a small region in aneighborhood of B Θ T containing the desired point of focus (see Figure 1.2). We relate the above theorem tofocusing via a corollary at the end of this section. Remark.
As mentioned in [14] as well, this result only describes how to control multiple scattering of thepressure field, but says nothing about the velocity field at time T ; hence energy is not controlled and thewave field may still have a large kinetic energy even at time T . Also, after the time t “ T , the Cauchy datainside Θ ‹ T generate waves that may and generally do enter the inner layer Θ T even before time t “ T .The main advantage of scattering control is that it controls both the pressure and velocity field so that for T ď t ď T , the wave generated by the time T Cauchy data inside Θ ˚ T will not penetrate the domain ofinfluence of the direct transmission ¯ π T R T r . Proof.
We start with (i). Suppose we found a wave field u such that u p T q has support in Θ T , and Cauchydata r “ r ` K tail as in the statement of the theorem. Let us denote w p t q “ u p T ` t q ` u p T ´ t q . Observe that w p q “ outside Θ T , and w t p q “ . By finite propagation speed, one also has w p t, x q “ when d p x, Θ T q ą t . Notice that all points in Θ ‹ are atleast distance T away from Θ T so one has π ‹ w p T q “ This precisely means that u p T q “ ´ u p q on Θ ‹ and ´ u t p T q “ ´ u t p q on Θ ‹ . Written in operator form, this amounts to π ‹ ν ˝ R T r “ ´ π ‹ r , where we recall that R s does not just propagate s units of time, but also give the Cauchy data at time t “ s .Plugging in r “ r ` K tail above gives π ‹ R p r ` K tail q “ ´ π ‹ p r ` K tail qô π ‹ Rr ` π ‹ RK tail “ ´ π ‹ r ´ π ‹ K tail “ ´ K tail ô K tail ` π ‹ RK tail “ ´ π ‹ Rr . (5.4)34 roof of (ii) First, if one adds r to both side of (5.4), and brings ´ π ˚ Rr to the the left hand side, oneobtains p I ` π ˚ R q r “ r . (5.5)Again denote u p t q “ p F r qp t q , and let w p t q be a superposition of u p t q and its time reversal; that is w p t q “ p F r qp t q ` p F r qp T ´ t q . Then using (5.5) and recalling that r vanishes outside of Θ , we have w p q “ r ` Rr is harmonic in Θ ‹ . Similarly, w p T q “ R T r ` ν ˝ r “ ν ˝ p Rr ` r q is harmonic in Θ ‹ . Note that w t p T q “ “ w t p q in Θ ‹ . Since w also solves that wave equation, then B t w vanishes wherever w is harmonic. By translation invariance of the wave operator, B t w (the mollification argument to make thisprecise is exactly as in the proof of (2.13)) also solves the wave equation while also having Cauchy dataat times t “ and t “ T vanishing in Θ ˚ . By Lemma 3, B t w p T q “ inside Θ ‹ T . Looking at just the firstcomponent of w p T q this says exactly that u p T q is harmonic in Θ ‹ T , which is equivalent to Π ‹ T u p T q “ . Thesecond statement in the theorem follows from finite propagation speed, as K tail is supported in Θ ‹ . Proof of (iii)
Suppose that K tail ` π ‹ RK tail “ . Since π ‹ is a projection and R is unitary, one has (cid:107) π ‹ RK tail (cid:107) ď (cid:107) K tail (cid:107) . However, since K tail “ ´ π ‹ RK tail , then the inequality above must in fact be an equality and so (cid:107) π ‹ RK tail (cid:107) “ (cid:107) K tail (cid:107) . Since R is unitary, one has (cid:107) K tail (cid:107) “ (cid:107) RK tail (cid:107) “ (cid:107) π ˚ RK tail (cid:107) ` (cid:107) ¯ πRK tail (cid:107) “ (cid:107) K tail (cid:107) ` (cid:107) ¯ πRK tail (cid:107) . Thus, ¯ πRK tail “ and so K tail “ ´ π ‹ RK tail “ ´ RK tail , implying that K tail P G . Proof of (iv)
Denote K l “ ř lj “ p´ π ‹ R q j p´ π ‹ Rr q . The proof follows almost verbatim as the proof showingthe density of the set Q defined in (2.19).In order to make Remark 5.2 more transparent on how this theorem relates to focusing, we add thefollowing corollary. First, we conjecture that following the methods of boundary control in [8], one mayextract certain travel times between points on the boundary to points in the interior and use that to create an r supported outside Ω , such that at a time T , the first component of R T p r q| Ω T has singular support equalto a single point. Thus we believe that it will be possible to satisfy the assumption in the following corollaryusing boundary control methods. Corollary 5.2.
Suppose r P C , a time t “ T , and Θ Ą Ω are such that supp p r q Ă Θ and the singular support of F p r qp T q| Θ T is nontrivial, contained inside B (cid:15) p x f q for some small (cid:15) ą . Then if K tail solves (5.4), then the singularsupport of u p T q is nontrivial and contained in B (cid:15) p x f q . The corollary is stated using the energy spaces employed throughout the paper. However, we believe itcan be refined to encompass general distributions and in particular a point singular support so that one hasa focusing wave in the usual sense.
Remark.
We emphasize again that despite the attractiveness of the corollary, it only gives focusing of thepressure field and says nothing about the velocity field. Thus, once one goes past time t “ T , one has lost allcontrol and one has no information on the wave field at such times, which is usually quite complex since K tail needs to be quite complicated in order to control the multiple scattering that allows focusing. Thus, thescattering control procedure is much more useful in this regard.35e close this section with an analogous theorem to Theorem 5.1 which controls the multiple scattering ofthe velocity field instead. The proof is almost identical excepting sign changes so we omit it. Theorem 5.3. (Multiple scattering control of velocity field) Let u be the solution to the wave equation with Cauchydata r “ r ` K tail P C , where r has support in Θ , and K tail has support outside Θ . Let T ą .(i) (Necessity) If u t p T q has support in Θ T , then necessarily K tail satisfies the following equation K tail ´ π ‹ RK tail “ ´ π ‹ Rr (ii) (Partial converse) Suppose K tail satisfies K tail ´ π ‹ RK tail “ ´ π ‹ Rr . Then u t p T q| Θ ‹ T “ and u p T q| Θ T “ R T r | Θ T . (iii) (Uniqueness of the tail) Any two tails may only differ by Cauchy data that is totally internally reflected, anddoes not penetrate Θ in time T . That is, if K tail ´ π ‹ RK tail “ , then K tail “ in C .(iv) (Almost Solvability) The set of r P H for which one has a convergent Neumann series solution for K tail , Q : “ t r P H : p I ´ π ‹ R q ´ r P C u is dense in H .Remark. We note that an almost identical proof used to recover kinetic energy of the almost direct transmissionin Proposition 2.8 and 2.9 may be used here to recover this energy from K tail instead.At this point, one might be led to believe that information may be lost or gained by using our Cauchydata setup versus the boundary setup that is done in Rose. This is actually not the case, and we show in thenext section that in one dimension, where one does not worry about glancing rays, both formulations arecompletely equivalent. For simplicity, we assume here that Ω occupies x ą and Θ is exactly the half-space t x ą ´ (cid:15) u for some (cid:15) ą .Without loss of generality, we assume that the wave speed is constantly equal to outside Ω , i.e. c | Ω ‹ “ .Then any wave field inside Ω ‹ is of the form u | Ω ‹ “ f p t ´ x q ` g p x ` t q (5.6)We assume that supp p f p s qq Ă t´ T ă s ă T ` (cid:15) u ( T is the focusing time; i.e. we are focusing at a point x T which is distance T away from using the metric determined by c ) and that the left going wave g is activatedonly after the right going wave f hits the boundary t x “ u . Precisely, this means that supp p g p s qq Ă t s ą ´ T u . As described in the last section, one has g p t q “ R ˚ f “ ż R p t ´ t q f p t q dt . (5.7)This is well-defined in an exact sense precisely since there are no glancing rays in 1 space dimension. See forexample [1] for details.To avoid dealing with harmonic extensions, as they do not add anything essential, we will assume that R applied to any of our Cauchy data has trace on B Θ ˚ “ t x “ u . This merely ensures that π ˚ R “ Θ ˚ R “ t x ă´ (cid:15) u R g p s q “ when s ď and using the support condition of f , our Cauchy data(initially given at t “ ´ T as opposed to t “ ) and its time- T propagation is ˜ f p x q : “ u p´ T q “ ˆ f p´ T ´ x q f p´ T ´ x q ˙ ,π ˚ R p u p´ T qq “ π ˚ R ˜ f “ t x ă´ (cid:15) u ˆ g p T ` x q´ g p T ` x q ˙ . (5.8)Then by p . q we have p π ˚ R ˜ f qp t ´ T q “ t t ă T ´ (cid:15) u ν g p t q “ t t ă T ´ (cid:15) u ν p R ‹ f qp t q , (5.9)where we get an equation for g p t q by differentiating (5.7), and we use the notation f , g to represent a columnvector of f, g and their derivative. Let us denote J C (cid:1) B as the Cauchy-to-boundary map, which maps Cauchydata at time t “ ´ T to boundary data on t x “ u . In this simple setting, it is well-defined as a map J C (cid:1) B : D p R x q Ñ D p R t q explicitly defined on smooth functions as J C (cid:1) B v p t q “ v p t ´ T q with an obvious extension to elements in C . Since ˜ f “ J ´ C (cid:1) B f p´¨q and R ‹ φ p´¨q “ R φ p¨q , we have a nicerelationship between R T and R given by J C (cid:1) B R T J ´ C (cid:1) B p f p´¨qq “ R p f p´¨qq for t ă T. (5.10) Proposition 5.4. (Equivalence of Rose and Cauchy-Marchenko in one dimension) Let f p t q “ K tail p t q ` r p t q denotethe incoming boundary data, and ˜ f p x q “ J ´ C (cid:1) B p K tail p t q ` r p t qq : “ ˜ K tail p x q ` ˜ r p x q be the corresponding Cauchydata at time ´ T with all the assumptions described earlier. Then, ˜ K tail satisfies the Cauchy-Marchenko equation with ˜ r iff K tail satisfies the Rose equation with r ; that is, ˜ K tail p x q ` π ˚ R ˜ K tail p x q “ ´ π ˚ R ˜ r ô K tail p´ t q ` R p K tail p´¨qq “ ´ R p r p´¨qq for t ă T ´ (cid:15) Proof.
Suppose we start with the Cauchy-Marchenko equation in the form (5.5) (translating everything bytime T and using the notation of boldface letters to represent a vector consisting of the funcion and its timederivative): u p´ T q ` π ‹ R p u p´ T qq “ ¯ π u p´ T qô ˜ f p x q ` π ‹ R ˜ f p x q “ ¯ π ˜ f p x q (5.11) ô J C (cid:1) B ˜ f ` J C (cid:1) B π ‹ R ˜ f “ J C (cid:1) B ¯ π ˜ f (5.12) ô f p´ t q ` t t ă T ´ (cid:15) u ν p R ‹ f qp t q “ r p´ t q This is essentially the right equation for Rose, but we rewrite it in the more familiar form: f p´ t q ` t t ă T ´ (cid:15) u ν p R ‹ f qp t q “ r p´ t qô K tail p´ t q ` ν p R ‹ K tail qp t q “ ´ ν p R ‹ r qp t q for t ă T ´ (cid:15) ô K tail p´ t q ` ν R p K tail p´¨qq “ ´ ν R p r p´¨qq for t ă T ´ (cid:15), ô K tail p´ t q ` R p K tail p´¨qq “ ´ R p r p´¨qq ddt r K tail p´ t q ` R p K tail p´¨qqs “ ´ ddt R p r p´¨qq for t ă T ´ (cid:15) ô K tail p´ t q ` R p K tail p´¨qq “ ´ R p r p´¨qq for t ă T ´ (cid:15) where the first equality is obtained be subtracted r p´ t q from both sides of the first equation and writing f “ r ` K tail . 37 emark. The above result helps explain the truncation that Rose does in [14] to obtain his autofocusingalgorithm. The Corollary essentially shows that K tail p t q must satisfy t t ă T ´ (cid:15) u K tail p´ t q ` t t ă T ´ (cid:15) u R p K tail p´¨qq “ ´ t t ă T ´ (cid:15) u R p r p´¨qq One naturally assumes that the tail come after the direct transmission r , which means K tail p t q is supportedin t ą ´ T ` (cid:15) and hence t t ă T ´ (cid:15) u K tail p´ t q “ K tail p´ t q . Thus, the Neumann series becomes K tail p´ t q “ ´ t t ă T ´ (cid:15) u R p r p´¨qq ` p t t ă T ´ (cid:15) u R q p r p´¨qq´ p t t ă T ´ (cid:15) u R q p r p´¨qq ` . . . and we may clearly see the truncation happening at each step of the algorithm. The truncation is essentialsince we just proved the equivalence of Rose’s algorithm to our Cauchy scheme, and we already provedthat our equation (5.4) is necessary and sufficient to control multiple scattering. The proof shows that thetruncation essentially comes from (5.4) only holding within a certain region in space (i.e. Θ ‹ in that theorem)that was determined by finite speed of propagation and unique continuation. In one dimension and afterusing the Cauchy-to-Boundary map, this spatial region corresponds to the time-truncation appearing inRose.We will describe in the following sections the connection between the equations of the previous theorems,the Marchenko equation, and scattering control. Burridge [4] considers the 1-dimensional inverse scattering problem for the plasma wave operator l q “ l ` q p x q where q “ in x ă . (recall that in 1 dimension, the acoustic wave equation may be put into thisform by a change of variables as in [4]). Since it is not relevant for this part, we will avoid describing thefunction spaces where all of our distributions here belong. One is interested in solutions to l q u “ withcertain boundary conditions at x “ that allow for only left-going solutions inside x ă (see [4, Section 3]for details). It is shown in [4] that there is a special Green’s function solution of the form G “ δ p t ´ x q` K p x, t q such that supp p K q Ă t| t | ď x, x ě u and one may recover q from knowing K .The given data are the reflected waves due to a right-going incidence wave in the region x ă . Analyti-cally, there is a causal Green’s function: G p x, t q “ δ p t ´ x q ` K p x, t q with supp p K q Ă t t ě | x | , t ą u . One is given the data M p t q “ K p x “ , t q (interpreted as a generalizedtrace), and the goal is to recover K from R . Then it is shown in [4, Section 3] that for each fixed x , K mustsatisfy the following integral equation known as the Marchenko equation: K p x, t q ` ż x ´ x K p x, τ q M p t ` τ q dτ “ ´ M p t ` x q for t ă x. (5.13)To relate this to (5.3), change variables to travel time coordinates z “ ż x c p x q ´ dx . Comparing with (5.1), we see that t f “ z p x f q and K p z, t q “ Ω tail p´ t ; z q solves the Marchenko equation abovewith R as the given data in place of M . The connection to (5.3) is now readily apparent from the previoussubsections. Notice that the proof of multiple scattering control in Theorem 5.1 and its corollary essentially utilizes theoperators I ` π ‹ R and I ´ π ‹ R to control scattering from the pressure field and the velocity field respectively.38 T t xT a‘ ‘ (a)
A pulse with Rose’s tail T t xT (b)
A pulse with scattering control tail
Figure 5.1:
These figures correspond to the incident pulse in Figure 1.1a. In Rose’s setup, the tail has extra waves toensure the pressure field is quiescent exactly at t “ T except for the direct transmission. In (a), the tail (constructedby the formula in (5.3)) consists of three (positive amplitude) waves being sent in after the (positive amplitude)incident pulse. The first wave cancels a returning wave which would create further scattering between the interfaces.The other two waves in the tail cancel the backscattered (negative amplitude) waves at t “ T , and only there. Thus,at t “ T , the singular support of the pressure field is precisely one point determined by the direct transmission.Part (b) shows the tail constructed using the scattering control algorithm. For scattering control, we only care aboutthe returning bicharacteristics, so the tail consists of only one wave to eliminate the one returning wave. Thus, for t P r T, T s the total wave field only consists of the direct transmission and two waves that will never go deeperinto the medium. Our scattering control series is a middle ground that allows one to control scattering in both the pressurefield and the velocity field such that after time t “ T , the exterior data coming from the direct transmissionis distinguished. Indeed, the scattering control operator is precisely I ´ π ˚ Rπ ˚ R “ p I ´ π ˚ R qp I ` π ˚ R q , whose Neumann series solutions involve exactly the even terms in the Neumann series of I ´ π ‹ R . Figure 5.1depicts the differences between Rose’s autofocusing and scattering control in a simple one-dimensionalexample. A Wave equation parametrix with reflection and transmission
We briefly review how a parametrix for the acoustic wave equation initial value problem with piecewisesmooth wave speed may be constructed in terms of reflections and transmissions, neglecting glancingrays. This is now-classical FIO theory, drawing from the work of many authors, including Chazarain [6],Hansen [10], and Taylor [22]. As nothing novel is developed here, we do not include proofs; our goal issimply to provide a bookkeeping system for use in the paper.Recalling §3.1, consider c p x q piecewise smooth with singular support contained in disjoint closed smoothhypersurfaces Γ i , with Γ “ Ť Γ i . The interfaces separate R n z Γ into disjoint components Ω j . In order todistinguish the sides of each hypersurface Γ i , consider an exploded space Z in which the connected componentsof R n z Γ are separate. It may be defined in terms of its closure, as a disjoint union Z “ ğ j Ω j , Z “ ď j Ω j Ă Z. In this way, B Z contains two copies of each Γ i , one for each adjoining Ω j .39efore proceeding further, we perform a standard microlocal splitting in order to separate forward-and backward-moving singularities. Recall that B t ´ c ∆ factors microlocally into half-wave operators pB t ` iQ qpB t ´ iQ q . The full solution operator F is then equivalent microlocally to a sum of solution operators F ˘ corresponding to B t ˘ iQ , with initial data related by a microlocally invertible matrix Ψ DO P : F p f , f q ” F ` g ` ` F ´ g ´ , „ g ` g ´ ” P „ f f . (A.1)The Cauchy data p g ` , g ´ q may be interpreted as a single distribution g on a doubled space Z “ Z ` \ Z ´ containing two copies of Z .We now describe a parametrix ˜ R for R “ ν ˝ R T as a sum of graph FIO on Z built from sequences ofreflections and transmissions, along with operators propagating data from one boundary to another, orpropagating the initial data to boundary data. The key feature of the propagators is that waves reaching theboundary of a subdomain Ω j simply leave Ω j rather than reflecting. To handle reflections and refractions,we record the outgoing boundary data left by waves escaping Ω j and convert them to appropriate incomingboundary data on each side of the interface, which generate reflected and refracted waves. Cauchy Propagators: J C (cid:1) S , J C (cid:1) S ` , J C (cid:1) B We first develop a reflectionless solution operator J C (cid:1) S for theCauchy problem on Z . To begin, extend each restriction c j “ c ˇˇ Ω j to a smooth function on R n . Let E ˘ j bethe half-wave Lax parametrix associated to B t ˘ iQ , Q “ p´ c j ∆ q { . Each η P ˚ T ˚ Ω ˘ ,j is associated with aunique c j -bicharacteristic γ η p t q in ˚ T ˚ R n passing through η at t “ , which may escape and possibly re-enter Ω ˘ ,j as t Ñ ˘8 .To prevent re-entry of wavefronts, we introduce a pseudodifferential cutoff ϕ p t, ξ q , omitting some detailsfor brevity. Let t e ˘ , t r ˘ denote the first positive and negative escape and re-entry times; let ϕ p t, γ η p t qq beidentically one on r t e ´ , t e ` s and supported in p t r ´ , t r ` q . Modify ϕ on a small neighborhood of R ˆ ˚ T ˚ B Ω ˘ ,j (the glancing rays) to ensure it is smooth. Finally, let J C (cid:1) S be the restriction of ϕ p t, D x q ˝ E ˘ j to R ˆ Ω ˘ ,j ; thisis the desired reflectionless propagator.We also require a variant J C (cid:1) S ` of J C (cid:1) S in which waves travel only forward in time. For this replace ϕ with some ϕ ` supported in p t e ´ , t r ` q and equal to 1 on r , t e ` s . Restricting J C (cid:1) S ` to the boundary, we obtainthe Cauchy-to-boundary map J C (cid:1) B “ J C (cid:1) S ` ˇˇ R ˆB Z .It can be shown (cf. [6]) that J C (cid:1) S , J C (cid:1) S ` P I ´ { p Z (cid:1) R ˆ Z q , and J C (cid:1) B P I p Z (cid:1) R ˆ B Z q . As desired, J C (cid:1) S and J C (cid:1) S ` are parametrices: pB t ˘ iQ q J C (cid:1) S h, pB t ˘ iQ q J C (cid:1) S ` h ” for WF p h q lying in a set V Ă ˚ T ˚ Z whose bicharacteristics are sufficiently far from glancing. By a direct argument with oscillatory integralrepresentations, it can also be shown that J C (cid:1) B is elliptic at covectors in V whose bicharacteristics intersect B Z . The near-glancing covector set W of §3 is then ˚ T ˚ Z z V . Boundary Propagators
Outgoing solutions from boundary data f P D p R ˆ Z q may be obtained bymicrolocally converting boundary data to Cauchy data, then applying J C (cid:1) S . The boundary-to-Cauchyconversion can be achieved by applying a microlocal inverse of J C (cid:1) B , conjugated by the time-reflectingmap S s : t ÞÑ s ´ t for an appropriate s . More precisely, near any covector β “ p t, x ; τ, ξ q P B Ω ˘ ,j in thehyperbolic region | τ | ą c j | ξ | there exists a unique bicharacteristic γ passing through β and lying inside Ω ˘ ,j in some time interval r s, t q , s ă t . Then J B (cid:1) S may be defined as S s J C (cid:1) S J ´ C (cid:1) B S s microlocally near β .On the elliptic region | τ | ă c j | ξ | define J B (cid:1) S as a parametrix for the elliptic boundary value problem; seee.g. [17, §4.8]. Applying a microlocal partition of unity, we obtain a global definition of J B (cid:1) S away from aneighborhood of the glancing region | τ | “ c j | ξ | . It can be proven that J B (cid:1) S P I ´ { p R ˆ B Z (cid:1) R ˆ Z q . Itsrestriction to the boundary r B ˝ J B (cid:1) S consists of a pseudodifferential operator equal to the identity on W andan elliptic graph FIO J B (cid:1) B P I p R ˆ B Z (cid:1) R ˆ B Z q describing waves traveling from one boundary to another. Reflection and Transmission
It is well known that transmitted and reflected waves arise from requiring aweak solution to be C at interfaces. Given incoming boundary data f P E p R ˆ B Z q (an image of J C (cid:1) B or That is, p di q ˚ γ p t q “ β , where i : B Z ã Ñ Z . B (cid:1) B ) microsupported near β , we seek data f R , f T satisfying the C constraints f ` f R ” ιf T , B ν p υJ B (cid:1) S υf ` J B (cid:1) S f R q ˇˇ R ˆB Z ” ι B ν J B (cid:1) S f T ˇˇ R ˆB Z . (A.2)Here, υ is time-reversal, so υJ B (cid:1) S υ is the outgoing solution that generated f . The map ι : B Z Ñ B Z reversesthe copies of each boundary component within B Z , and B ν denotes the normal derivative. The secondequation in (A.2) simplifies to a pseudodifferential equation N I f ` N R f R ” N T f T (A.3)with operators N I , N R , N T P Ψ p R ˆ B Z q that may be explicitly computed. The system (A.2–A.3) maybe microlocally inverted to recover f R “ M R f , f T “ M T f in terms of pseudodifferential reflection andtransmission operators M R , ιM T P Ψ p R ˆ B Z q . Let M “ M R ` M T .The principal symbols of M R and ιM T have well-known geometric interpretations. In the doublyhyperbolic region where | τ | ă c | ξ | on both sides of the interface, σ p M R q “ cot θ R ´ cot θ T cot θ R ` cot θ T , σ p ιM T q “ θ R cot θ R ` cot θ T , (A.4)where θ R , θ T are the angles between the normal and the associated reflected and transmitted bicharacteristics.Here cot θ R “ ` c ´ R τ ´ | ξ | ˘ { { | ξ | , where c R is the wave speed at β on the reflected side, and similarly for θ T . From (A.4) we deduce M T is elliptic in the doubly-hyperbolic region, while M R is elliptic as long as c is discontinuous at the interface. Note that while the principal symbol of ιM T may exceed 1, this does notviolate energy conservation since M T operates on boundary rather than Cauchy data. Parametrix
With all the necessary components defined, we now set ˜ F “ J C (cid:1) S ` J B (cid:1) S M ÿ k “ p J B (cid:1) B M q k J C (cid:1) B , ˜ R “ r T ˝ ˜ F , (A.5)where r T is restriction to t “ T , plus time-reversal. Again omitting the proof, it can be shown that ˜ F ” F and ˜ R ” R away from glancing rays; that is, for initial data h such that every broken bicharacteristicoriginating in WF p h q is sufficiently far from glancing. Recalling that M “ M R ` M T , we may write ˜ R as asum of graph FIO indexed by sequences of reflections and transmissions: ˜ R “ ÿ s Pt R,T u k k ě ˜ R s , ˜ R pq “ r T J C (cid:1) S , ˜ R p s ,...,s k q “ r T J B (cid:1) S M s k J B (cid:1) B ¨ ¨ ¨ M s J B (cid:1) B M s J C (cid:1) B . (A.6)The solution operator ˜ F likewise decomposes into analogous components ˜ F s . Comparison with Layered Media Parametrices
The above construction is in fact the natural generalizationfrom the flat interface case of a layered media. Indeed, suppose our space Θ is only a small perturbation of theflat layered media case (see [23] for notation and analysis in the flat case). This ensures that bicharacteristicsegments starting from Γ i hit Γ i ´ or Γ i ` first before hitting another interface (here, Ω i lies below Γ i andabove Γ i ` ). The full wave field may be microlocally decomposed into upgoing and downgoing componentsat each interface Γ i denoted u p i q´ , resp. u p i q` as described in [20, proof of Theorem 3.1]. Then localizing theconstruction of the boundary-to-boundary maps J B (cid:1) B , we obtain J i,i ` B (cid:1) B (resp. J i,i ´ B (cid:1) B ), which propagate u i, ` (resp. u i, ´ ) to interface Γ i ` (resp. Γ i ´ ).Next, there are reflection and transmission operators, denoted R i,j , T i,j P Ψ p R ˆ Γ i q which are essentiallythe M R , M T operators from before but microlocally restricted to a particular “side” of a particular interface.41he indexing is such that R i,j denotes the reflection coefficient of a wave inside Ω j reflecting off of Γ i . While T i,j denotes the transmission coefficient for a wave from Ω i into Ω j where the constructions are made exactlyas in the previous section. Under this simplified geometry, the outgoing waves at interface Γ i are given by u p i q` “ T i ´ ,i J i ´ ,i B (cid:1) B u p i ´ q` ` R i,i J i ` ,i B (cid:1) B u p i ` q´ and u p i q´ “ R i,i ´ J i ´ ,i B (cid:1) B u p i ´ q` ` T i,i ´ J i ` ,i B (cid:1) B u p i ` q´ . This is all for i ě , while for i “ we must take into account the source term φ P D p Γ q (assuming thisis the only source) and only those incoming waves from Γ : u p q` “ R , J , B (cid:1) B u p q´ ` φ ` source u p q´ “ T , J , B (cid:1) B u p q´ ` φ ´ source . Denote u ˘ “ r u p q˘ , . . . , u p r q˘ s T . Thus, as done in [7], we may combine, the R, T operators and thecorresponding J B (cid:1) B occurring in the above formulas into one operator (for example, R i,i J i ` ,i B (cid:1) B becomes asingle operator). Then we form T ˘ and R ˘ , each a r ˆ r matrix of FIO’s, to obtain the following recursiveformula: „ u ` u ´ “ „ T ` R ` R ´ T ´ „ u ` u ´ ` „ p φ ` source , , . . . , q T p φ ´ source , , . . . , q T . Hence, it is fitting to denote S sc “ ” T ` R ` R ´ T ´ ı as the scattering “matrix”, which corresponds to J B (cid:1) B M appear-ing in (A.5). To connect this construction to (A.5), start with Cauchy data φ Cauchy P C with microsupportclose to a single covector, whose corresponding geodesic hits Γ transversely. Then the solution restricted to Γ near this first intersection is microlocally equal to φ Γ “ φ incoming ` φ source , where φ incoming “ J C (cid:1) B φ Cauchy and φ ` source “ T , J C (cid:1) B φ Cauchy and φ ´ source “ R , J C (cid:1) B φ Cauchy . So the upgoingand downgoing parts of the solution at the interfaces are given by „ u ` u ´ “ « p φ ` incoming , , . . . , q T p φ ´ incoming , , . . . , q T ff ` ÿ k “ S ksc „ p φ ` source , , . . . , q T p φ ´ source , , . . . , q T . After applying the boundary to solution operator, we obtain a formula exactly analogous to (A.5), and onecan use the scattering matrix to track the principal symbols of the wave field in each Ω i separately. Funding Acknowledgements:
P. C. and V. K. were supported by the Simons Foundation under the MATH ` X program. M. V. dH. was partially supported by the Simons Foundation under the MATH ` X program,the National Science Foundation under grant DMS-1559587, and by the members of the Geo-MathematicalGroup at Rice University. G. U. is Walker Family Endowed Professor of Mathematics at the University ofWashington, and was partially supported by the National Science Foundation, a Si-Yuan Professorship atHong Kong University of Science and Technology, and a FiDiPro Professorship at the Academy of Finland.
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