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Mathematics of Seismic ImagingPart 4: Global Asymptotics
William W. Symes
Rice University
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4. Global Asymptotics
Why Beylkin isn’t enough
Geometry of Reflection
Development of Global Asymptotics
Kinematic Artifacts in Migration
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4. Global Asymptotics
Why Beylkin isn’t enough
Geometry of Reflection
Development of Global Asymptotics
Kinematic Artifacts in Migration
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The theory developed by Beylkin and others cannotbe the end of the story:
B. White, “The Stochastic Caustic” (1982): For“random but smooth” v(x) with variance σ, pointsat distance O(σ−2/3) from source have more thanone ray connecting to source, with probability 1 -multipathing associated with formation of caustics= ray envelopes.
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Example: Marmousi, lightly smoothed
2
1
0
zHkmL
3 4 5 6 7 8 9x HkmL
5.5 km�s
1.5 km�s
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Example: Marmousi, lightly smoothed
2
1
0
zHkmL
3 4 5 6 7 8 9x HkmL
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Multipathing, formation of caustics invalidatesasymptotic analysis on which GRT representation isbased:
multiple rays ⇒ traveltime no longer function ofposition
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Why it matters
Strong refraction: salt(4-5 km/s) structuresembedded in sedimentaryrock (2-3 km/s) (eg. Gulfof Mexico), also chalk inNorth Sea, gas seeps -some of the mostpromising petroleumprovinces!
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4. Global Asymptotics
Why Beylkin isn’t enough
Geometry of Reflection
Development of Global Asymptotics
Kinematic Artifacts in Migration
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Escape from simplicity - the CanonicalRelation
How do we get away from “simple geometricoptics”, SSR, DSR,... - all violated in sufficientlycomplex (and realistic) models?
Rakesh Comm. PDE 1988, Nolan Comm. PDE1997: global description of Fδ[v ] as mappingreflectors 7→ reflections.
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Escape from simplicity - the CanonicalRelation
Y = {xs , t, xr} (time × set of source-receiver pairs)submfd of R7 of dim. ≤ 5, Π : T ∗(R7)→ T ∗Y thenatural projection
supp r ⊂ X ⊂ R3
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Escape from simplicity - the CanonicalRelation
Canonical relationCFδ[v ] ⊂ T ∗(X ) \ {0} × T ∗(Y ) \ {0} describessingularity mapping properties of F :
(x, ξ, y, η) ∈ CFδ[v ] ⇔ for some u ∈ E ′(X ),
(x, ξ) ∈ WF (u)and (y, η) ∈ WF (Fu)
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Rakesh’s Construction
recall defn of rays: solutions of Hamiltonian system
dX
dt= ∇ΞH(X,Ξ),
dΞ
dt= −∇XH(X,Ξ)
with
H(X,Ξ) =1
2(1− v 2(X)|Ξ|2) = 0
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Rakesh’s Construction
Characterization of CF :
((x, ξ), (xs , t, xr , ξs, τ, ξr)) ∈ CFδ[v ]
⊂ T ∗(X )− {0} × T ∗(Y )− {0}⇔ there are rays (Xs ,Ξs), (Xr ,Ξr), times ts , tr sothat
Π(Xs(0), t,Xr(t),Ξs(0), τ,Ξr(t)) = (xs , t, xr , ξs , τ, ξr),
Xs(ts) = Xr(t−tr) = x, ts+tr = t, Ξs(ts)−Ξr(t−tr)||ξ
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Rakesh’s Construction
|Ξs(ts)| = |Ξr(t − tr)| ⇒
sum = bisector ⇒
velocity vectors of incident ray from source andreflected ray from receiver (traced backwards intime) make equal angles with reflector at x withnormal ξ.
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Rakesh’s Construction
(t−t )
X
sΞ (t )
s
X s
x
x x ξ rs, r,sξ,
Ξ s
ξ,
rΞr,
,
ΠΠ
−Ξr r
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Rakesh’s Construction
Upshot: canonical relation of Fδ[v ] simply enforcesthe equal-angles law of reflection.
Further, rays carry high-frequency energy, in exactlythe fashion that seismologists imagine.
Finally, Rakesh’s characterization of CF is global: noassumptions about ray geometry, other than noforward scattering and no grazing incidence on theacquisition surface Y , are needed.
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Rakesh’s Construction
Inversion aperture Γ:
(x, ξ) ∈ Γ ⇔
there is (xs , t, xr) ∈ Y and rays connecting xs andxr with x so that ξ bisects ray velocity vectors, andtotal time along the two rays is = t
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Rakesh’s Construction
(t−t )
X
sΞ (t )
s
X s
x
x x ξ rs, r,sξ,
Ξ s
ξ,
rΞr,
,
ΠΠ
−Ξr r
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Rakesh’s ConstructionExercise based on HorizontalReflector2D.py
Explain why inverted layers do not extend toboundary - explain where they end
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Proof: Plan of attack
Recall that
F [v ]r(xr , t; xs) =∂δu
∂t(xr , t; xs)
where1
v 2∂2δu
∂t2−∇2δu =
1
v 2∂2u
∂t2r
1
v 2∂2u
∂t2−∇2u = δ(t)δ(x− xs)
and u, δu ≡ 0, t < 0.
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Proof: Plan of attack
Need to understand (1) WF (u), (2) relationWF (r)↔ WF (ru), (3) WF of soln of WE in termsof WF of RHS (this also gives (1)!).
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Singularities of the Acoustic PotentialField
Main tool: Propagation of Singularities theoremof Hormander (1970).
Given symbol p(x, ξ) w. asymptotic expansion,order m, define null bicharateristics (= rays) assolutions (x(t), ξ(t)) of Hamiltonian system
dx
dt=∂pm∂ξ
(x, ξ),dξ
dt= −∂pm
∂x(x, ξ)
with pm(x(t), ξ(t)) ≡ 0.
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Singularities of the Acoustic PotentialField
Theorem: Suppose p(x,D)u = f , t 7→ (x(t), ξ(t))is a null bicharacteristic, and for t0 ≤ t ≤ t1,(x(t), ξ(t)) /∈ WF (f ). Then either
I {(x(t), ξ(t)) : t0 ≤ t ≤ t1} ⊂ WF (u)
or
I {(x(t), ξ(t)) : t0 ≤ t ≤ t1} ⊂ T ∗(Rn)−WF (u)
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Source to Field
RHS of wave equation for u = δ function in x, t.WF set = {(x, t, ξ, τ) : x = xs , t = 0} - i.e. norestriction on covector part.
⇒ (x, t, ξ, τ) ∈ WF (u) iff it lies on a null bicharpassing over (xs , 0)
⇒ (x, t) lies on the “light cone” with vertex at(xs , 0).
[Principal symbol for wave op isp2(x, t, ξ, τ) = 1
2(τ 2 − v 2(x)|ξ|2)]
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Source to Field
Hamilton’s equations for null bicharacteristics (usingt for parameter) are
dt
dt= 1 = τ,
dτ
dt= 0
dX
dt= −v 2(X)Ξ,
dΞ
dt= ∇ log v(X)
Thus ξ is proportional to velocity vector of ray.
[Exercise: show (ξ, τ) normal to light cone]
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Singularities of ProductsTo compute WF (ru) from WF (r) and WF (u), useGabor calculus (Duistermaat, Ch. 1)
r is really (r ◦ π), where π(x, t) = x. Choose bumpfunction φ localized near (x, t)
[φ(r◦π)u]∧(ξ, τ) =
∫dξ′ dτ ′φr(ξ′)δ(τ ′)u(ξ−ξ′, τ−τ ′)
=
∫dξ′φr(ξ′)u(ξ − ξ′, τ)
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decays rapidly as |(ξ, τ)| → ∞ unless (i) exist(x′, ξ′) ∈ WF (r) so that x, x′ ∈ π(suppφ),(ξ − ξ′, ·) ∈ WF (u), i.e.(ξ, ·) ∈ WF (r ◦ π) + WF (u), or (ii) ξ ∈ WF (r) or(ξ, ·) ∈ WF (u).
Possibility (ii) will not contribute, so effectively
WF ((r ◦ π)u) = {(x, ts , ξ + Ξs(ts), ·) :
(x, ξ) ∈ WF (r), x = Xs(ts)}
for a ray (Xs ,Ξs) with Xs(0) = xs , some τ .
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Wavefront set of Scattered FieldPropagation of singularities:(xr , t, ξr , τr) ∈ WF (δu)⇔ on ray (Xr ,Ξr) passingthrough WF (ru). Can argue that time ofintersection is t − tr < t (Exercise: do it!)
That is,
Xr(t) = xr ,Xr(t − tr) = Xs(ts) = x ,
t = tr + ts , and
Ξr(ts) = ξ + Ξs(ts)
for some ξ ∈ WF (r). Q. E. D.
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4. Global Asymptotics
Why Beylkin isn’t enough
Geometry of Reflection
Development of Global Asymptotics
Kinematic Artifacts in Migration
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Rakesh’s Thesis
Rakesh (1986):
I F [v ] is Fourier Integral Operator = class ofoscillatory integral operators, introduced byHormander and others in the ’70s to describethe solutions of nonelliptic PDEs (ΨDOs arespecial FIOs.)
I Adjoint of FIO = FIO with inverse canonicalrelation
I Composition of FIOs 6= FIO in general - not analgebra (unlike ΨDOs)
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Rakesh’s Thesis
I Beylkin: F [v ]∗F [v ] is FIO (ΨDO, actually),given simple ray geometry hypothesis - but thisis only sufficient
I Rakesh: follows from general results ofHormander: simple ray geometry ⇔ canonicalrelation is graph of ext. deriv. of phasefunction.
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The Shell Guys and TIC
Inversion aperture 6= T ∗X ⇒ F [v ]∗F [v ] cannot beboundedly invertible
[Exercise: Why? Hint: revisit relation betweensymbol and operator, recall that inversion apertureis not all of T ∗X ]
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The Shell Guys and TIC
A microlocal parametrix for a ΨDO P in a conic setΓ is an operator Q for which
u − QPu ∈ C∞
if WF (u) ⊂ Γ
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The Shell Guys and TIC
Smit, tenKroode and Verdel (1998): provided that
I source, receiver positions (xs , xr) form an open4D manifold (“complete coverage” - all source,receiver positions at least locally), and
I the Traveltime Injectivity Condition (“TIC”)holds: C−1F [v ] ⊂ T ∗Y \ {0} × T ∗X \ {0} is a
function - that is, initial data for source andreceiver rays projected into T ∗Y and totaltravel time together determine ray pairuniquely....
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The Shell Guys and TIC
then F [v ]∗F [v ] is ΨDO
⇒ application of F [v ]∗ produces image
and F [v ]∗F [v ] has microlocal parametrix(“asymptotic inverse”) in inversion aperture
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TIC is a nontrivial constraint!
x x
x xs r
Symmetric waveguide: time (xs → x→ xr) same astime (xs → x→ xr), so TIC fails.
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Stolk’s Thesis
Stolk (2000): for dim=2, under “completecoverage” hypothesis, v for which F [v ]∗F [v ] =[ΨDO + rel. smoothing op] open, dense set inC∞(R2) (without assuming TIC!). Conjecture:same for dim=3.
Also, for any dim, v for which F [v ]∗F [v ] is FIOopen, dense in C∞(R2).
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Operto’s Thesis
Application of F [v ]∗ involves accounting for all raysconnecting source and receiver with reflectors.
Standard practice at time attempted to simplifyintegral kernel with single choice of ray pair(shortest time, max energy,...).
Operto et al (2000): nice illustration that all raysmust be included in general to obtain good image.
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Nolan’s Thesis
Limitation of Smit-tenKroode-Verdel: mostidealized data acquisition geometries violate“complete coverage”: for example, idealized marinestreamer geometry (src-recvr submfd is 3D)
Nolan (1997): result remains true without“complete coverage” condition: requires only TICplus addl condition so that projection CF [v ] → T ∗Yis embedding - but examples violating TIC are mucheasier to construct when source-receiver submfd haspositive codim.
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Nolan’s Thesis
Sinister Implication: When data is just a singlegather - common shot, common offset - image maycontain artifacts, i.e. spurious reflectors not presentin model.
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4. Global Asymptotics
Why Beylkin isn’t enough
Geometry of Reflection
Development of Global Asymptotics
Kinematic Artifacts in Migration
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Horrible Example
Synthetic 2D Example (see Stolk and WWS,Geophysics 2004 for this and other horrible expls)
Strongly refracting acoustic lens (v) over horizontalreflector (r), Sobs = F [v ]r .
(i) for open source-receiver set, F [v ]∗Sobs = goodimage of reflector - within limits of finite frequencyimplied by numerical method, F [v ]∗F [v ] acts likeΨDO;
(ii) for common offset submfd (codim 1), TIC isviolated and WF (F [v ]∗Sobs) is larger than WF (r).
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2
1
0
x2
-1 0 1x1
1
0.6
Gaussian lens velocity model, flat reflector at depth2 km, overlain with rays and wavefronts (Stolk & S.2002 SEG).
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5
4
timeHsL
-2 -1 0 1 2receiver position HkmL
Typical shot gather - lots of arrivals
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1.6
2.0
2.4
x2
0.0 0.4 0.8x1
Image from common offset gather, h = 0.3 - TICfails (3 ray pairs with same data), image has“artifact” WF
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1.6
1.8
2.0
2.2
2.4
x2
0 0.5 1.0x1
Image from all offsets - TIC holds, “WF” recovered
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What it all means
Note that a gather scheme makes the scatteringoperator block-diagonal: for example with datasorted into common offset gathers h = (xr − xs)/2,
F [v ] = [Fh1[v ], ...,FhN [v ]]T , d = [dh1, ..., dhN ]T
Thus F [v ]∗d =∑
i Fhi [v ]∗dhi . Otherwise put: toform image, migrate ith gather (apply migrationoperator Fhi [v ]∗, then stack individual migratedimages.
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What it all means
Horrible Examples show that individual offset gatherimages may contain nonphysical apparent reflectors(artifacts).
Smit-tenKroode-Verdel, Nolan, Stolk: if TIC holds,then these artifacts are not stationary with respectto the gather parameter, hence stack out (interferedestructively) in final image.