Reverse Time Shot-Geophone Migrationand
velocity analysis, linearizing, inverse problems, extended models,etc. etc.
William W. Symes
PIMS Geophysical Inversion Workshop
Calgary, Alberta
July 2003
www.trip.caam.rice.edu
1
Agenda:
1. Partially linearized seismic inverse problem - the working en-
vironment of seismic processing
2. Velocity Analysis = solution method for partially linearized
inverse problem, based on invertible extended model
3. Strong lateral velocity variations ⇒ the usual extended mod-
els of MVA aren’t invertible
4. Extended model of shot-geophone migration is invertible,
even with strong lateral velocity variations!
5. Reverse time algorithm for shot-geophone migration
2
1. Partially linearized seismic inverse problem - the working en-
vironment of seismic processing
2. Velocity Analysis = solution method for partially linearized
inverse problem, based on invertible extended model
3. Strong lateral velocity variations ⇒ the usual extended mod-
els of MVA aren’t invertible
4. Extended model of shot-geophone migration is invertible,
even with strong lateral velocity variations!
5. Reverse time algorithm for shot-geophone migration
3
Simple acoustic model of seismic reflection inverse problem:
given data dobs, find velocity field c (a function on the subsurface
X) so that
F[c] = dobs
F = forward map aka modeling operator aka ..., defined by
F[c](xr,xs, t) =∂u
∂t(xr, t;xs),
where acoustic potential field u(x, t;xs) solves(1
c2∂2
∂t2−∇2
)u = f(t)δ(x− xs)
4
This problem is too hard!
Most useful progress based on these beliefs:
• If c = v(1 + r) where v is smooth and r oscillatory (or even
singular), then F[c] ' F[v] + DF[v](vr) (i.e. the Born ap-
proximation is accurate);
• In lots of places, the actual compressional velocity field in
the Earth has this nature.
[A good math problem: exactly how is the first bullet true? Lots
of numerical evidence, little mathematics.]
5
Partially linearized seismic inverse problem: given observed
seismic data dobs, find smooth velocity v ∈ E(X), X ⊂ R3 oscil-
latory reflectivity r ∈ E ′(X) so that
DF[v](vr) = F [v]r ' dobs
where the acoustic potential field u and its perturbation δu solve(1
v2
∂2
∂t2−∇2
)u = f(t)δ(x− xs),
(1
v2
∂2
∂t2−∇2
)δu = 2r∇2u
plus suitable bdry and initial conditions.
6
F [v]r =∂δu
∂t
∣∣∣∣Y
data acquisition manifold Y = {(xr, t;xs)} ⊂ R7, dimn Y ≤ 5
(many idealizations here!).
F [v] : E ′(X) → D′(Y ) is a linear map (FIO of order 1) (Rakesh
1988), but dependence on v is quite nonlinear, so this inverse
problem is nonlinear.
7
1. Partially linearized seismic inverse problem - the working en-
vironment of seismic processing
2. Velocity Analysis = solution method for partially linearized
inverse problem, based on invertible extended model
3. Strong lateral velocity variations ⇒ the usual extended mod-
els of MVA aren’t invertible
4. Extended model of shot-geophone migration is invertible,
even with strong lateral velocity variations!
5. Reverse time algorithm for shot-geophone migration
8
Extension of F [v] (aka extended model): manifold X and maps
χ : E ′(X) → E ′(X), F [v] : E ′(X) → D′(Y ) so that
F [v]E ′(X) → D′(Y )
χ ↑ ↑ idE ′(X) → D′(Y )
F [v]
commutes, i.e.
F [v]χr = F [v]r
9
(Familiar) Example: the Convolutional Model
• Approximation of P. L. model, accurate when v, r functions
of z only
• data function of t, h = (xr − xs)/2 half-offset
• two-way traveltime τ(z, h), inverse ζ(t, h)
• if v =const. then τ(z, h) = 2√
z2 + h2/v
F [v]r(t, h) =∫
dt′ f(t− t′)r(ζ(t′, h))
(”inverse NMO, convolve with source”)
10
0
0.2
0.4
0.6
0.8
1.0
t0 (
s)
1
,
0
0.2
0.4
0.6
0.8
1.0
t (s
)
0 1h (km)
,
0
0.2
0.4
0.6
0.8
1.0
t (s
)
0 1h (km)
Left: r(t0). Middle: r(ζ(t, h)). Right: F [v]r(t, h)
11
Factor convolutional model through extension: replicate r for
each h
χ : r(z) 7→ r(z, h) = r(z)
then apply inverse NMO and convolve with source, independently
for each h
F [v] : r(z, h) 7→ f ∗ r(ζ(t, h), h)
12
0
0.2
0.4
0.6
0.8
1.0
t0 (
s)
1
,
0
0.2
0.4
0.6
0.8
1.0
t0 (
s)
0 1h (km)
,
0
0.2
0.4
0.6
0.8
1.0
t (s
)
0 1h (km)
Left: r(t0). Middle: r(t0, h) = χr. Right: r(ζ(t, h), h).
13
Invertible extension: F [v] has a right parametrix G[v], i.e.
I − F [v]G[v]
is smoothing.
Example: for the convolutional model, G[v] is signature decon
followed by NMO, applied trace-by-trace.
NB: The trivial extension - X = X, F = F - is virtually never
invertible.
14
0
0.2
0.4
0.6
0.8
1.0
t (s
)
0 1h (km)
,
0
0.2
0.4
0.6
0.8
1.0
t0 (
s)
0 1h (km)
,
0
0.2
0.4
0.6
0.8
1.0
t0 (
s)
0 1h (km)
Left: d(t, h) = F [v]r(t, h). Middle: G[v]d(t0, h). Right:
G[v1]d(t0, h), v1 6= v
15
Reformulation of inverse problem: given dobs, find v so that
G[v]dobs ∈ the range of χ.
Non-Gary Proof: that is, G[v]dobs = χr for some r, so dobs 'F [v]G[v]dobs = F [v]χr = F [v]r Q. E. D.
This is velocity analysis!
Example: Standard VA. Apply convolutional model to each mid-
point in CMP-binned data. Range of χ = r(z, h) independent of
h, i.e. flat. SO: twiddle v so that G[v]dobs shows flat events.
Caveats: in practice, be happy when G[v]dobs is in range of χ ex-
cept for wrong amplitudes, finite frequency effects, and obvious
(!) noise.
16
0.6
0.8
1.0
1.2
1.4
1.6
1.8
time
(s)
10 20 30 40 50channel
,
0
0.5
1.0
1.5
2.0
time
(s)
0.5 1.0 1.5 2.0 2.5offset (km)
Left: part of survey (dobs) from North Sea (thanks: Shell Re-search), lightly preprocessed.Right: restriction of G[v]dobs to xm = const (function of depth,offset): shows rel. sm’ness in h (offset) for properly chosen v.
17
1. Partially linearized seismic inverse problem - the working en-
vironment of seismic processing
2. Velocity Analysis = solution method for partially linearized
inverse problem, based on invertible extended model
3. Strong lateral velocity variations ⇒ the usual extensions of
MVA aren’t invertible
4. Extended model of shot-geophone migration is invertible,
even with strong lateral velocity variations!
5. Reverse time algorithm for shot-geophone migration
18
The usual extended model behind Migration Velocity Analysis:
• v, r functions of all space variables
• χr(x,h) = r(x) (so r ∈ range of χ ⇔ plots of r(·, ·, z, h) appearflat)
•
F [v]r(xr,xs, t) =∂2
∂t2
∫dx r(x,h)
∫ds G(xr, t− s;x)u(xs, s;x)
(recall h = (xr − xs)/2)
NB: F is “block diagonal” - family of operators (FIOs) parametrizedby h.
19
• Beylkin (1985), Rakesh (1988): if ‖∇2v‖C0 “not too big”,
then the usual extension is invertible.
• G = common offset migration-inversion aka ray-Born inver-
sion aka true-amplitude migration etc. etc. Usually imple-
mented as generalized Radon transform = ”weighted diffrac-
tion stack” (Beylkin, Bleistein, DeHoop,...)
• Nolan, Stolk, WWS: if ‖∇2v‖C0 is too big, usual extension is
not invertible!
20
2
1
0
x2
-1 0 1x1
1
0.6
Example: Gaussian lens over flat reflector at depth 2 km (r(x) =
δ(z − 2)).
21
1.6
2.0
2.4
x2
0 1 2offset
3,2
3,1
1,2
2,1
3,3
1,1
travel time # s,r
Common Image Gather (const. x, y slice) of G[v]dobs: not flat,
i.e. not in range of χ even allowing for amplitude, finite
frequency errors, and even though velocity is correct! [Stolk,
WWS 2002]
22
1. Partially linearized seismic inverse problem - the working en-
vironment of seismic processing
2. Velocity Analysis = solution method for partially linearized
inverse problem, based on invertible extended model
3. Strong lateral velocity variations ⇒ the usual extensions of
MVA aren’t invertible
4. Extended model of shot-geophone migration is invertible,
even with strong lateral velocity variations!
5. Reverse time algorithm for shot-geophone migration
23
Claerbout’s extended model = basis of survey sinking or shot-geophone migration:
• χr(x,h) = r(x)δ(h), so r ∈ range of χ ⇔ plots of r(·, ·, z, h)appear focussed at h = 0
F [v]r(xr,xs, t)
=∂2
∂t2
∫dx
∫dh r(x,h)
∫ds G(xr, t− s;x + h)u(xs, s;x− h)
• This extension is invertible, assuming (i) h3 = 0 (horizontaloffset only) and (ii) ”DSR hypothesis”: rays do not turn.Then adjoint map is equivalent modulo elliptic ΨDO fac-tor to shot-geophone migration via DSR equation [Stolk-DeHoop 2001]
24
Lens data, shot-geophone migration [B. Biondi, 2002]
Left: Image via DSR. Middle: G[v]d - well-focused (at h = 0),
i.e. in range of χ to extent possible. Right: Angle CIG.
25
1. Partially linearized seismic inverse problem - the working en-
vironment of seismic processing
2. Velocity Analysis = solution method for partially linearized
inverse problem, based on invertible extended model
3. Strong lateral velocity variations ⇒ the usual extensions of
MVA aren’t invertible
4. Extended model of shot-geophone migration is invertible,
even with strong lateral velocity variations!
5. Reverse time algorithm for shot-geophone migration
26
Alternate expression for extended S-G model:
F [v]r(xr, t;xs) =∂
∂tδu(x, t;xs)|x=xr
where
(1
v(x)2∂2
∂t2−∇2
x
)δu(x, t;xs) =
∫x+2Σd
dy r(x,y)u(y, t;xs)
Computing G[v]: instead of parametrix, be satisfied with adjoint -
the two differ by a ΨDO factor, which will not affect smoothness
of CIGs.
27
Computing the adjoint: use the adjoint state method (WWS,Biondi & Shan, SEG 2002).
Define adjoint state w:(1
v(x)2∂2
∂t2−∇2
x
)w(x, t;xs) =
∫dxr d(xr, t;xs)δ(x− xr)
with w(x, t;xs) = 0, t >> 0. [This is exactly the backpropagatedfield of standard reverse time prestack migration, cf. Lines talk.]
Then
F [v]∗d(x,h) =∫
dxs
∫dt u(x + 2h, t;xs)w(x, t;xs)
[This is exactly the same computation as for standard reversetime prestack, except that crosscorrelation occurs at an offset2h].
28
0
0.1
0.2
0.3
0.4
0.5
de
pth
(km
)
-0.2 -0.1 0 0.1offset (km)
Offset Image Gather, x=1 km
,
0
0.1
0.2
0.3
0.4
0.5
de
pth
(km
)
-0.2 -0.1 0 0.1offset (km)
OIG, x=1 km: vel 10% high
,
0
0.1
0.2
0.3
0.4
0.5
de
pth
(km
)
-0.2 -0.1 0 0.1offset (km)
OIG, x=1 km: vel 10% low
Two way reverse time horizontal offset S-G image gathers of
data from random reflectivity, constant velocity. From left to
right: correct velocity, 10% high, 10% low.
29
Some Loose Ends
• invertibility of S-G extended model only known under DSRassumption with horizontal offsets [Stolk-DeHoop, 2001].Vertical offsets are good when DSR breaks down, eg. toimage overhanging reflectors [Biondi, WWS 2002]. Currentbest result: data focusses only at offset = 0 within a lim-ited range off offsets; focussing at large offsets not ruled out[WWS, 2002]. What actually happens?
• S-G extension amounts to construction of annihilators [cf.DeHoop]. How can one characterize globally invertible anni-hilator representations?
• quantification of non-membership in range of χ (DSO) -which ones yield good optimization problems locally [Stolk-WWS, IP 2003] or globally?
30
Conclusions
• Most of contemporary SDP related partially linearized seismicinverse problem
• Velocity analysis = approach to solution of PL seismic inverseproblem via invertible extended models
• Usual extended models (common offset, common shot, com-mon angle,...) are not invertible when the velocity structureis complex, due to multipathing
• The extended model of shot-geophone migration is invertibleeven in the presence of multipathing
• Shot-geophone migration has a reverse-time implementation
31
Thanks to:
• Chris Stolk, Biondo Biondi, Marteen DeHoop
• NSF, DoE
• the sponsors of The Rice Inversion Project
32
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