fwi + mva - rice university
TRANSCRIPT
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FWI + MVA
William Symes
The Rice Inversion Project
October 2012
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Agenda
FWI
Extended Modeling
Extended Modeling and WEMVA
Extended Modeling and FWI
Nonlinear WEMVA and LF Control
Summary
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M = model space, D = data space
F :M→D modeling operator = forward map =...
Full Waveform Inversion problem:
given d ∈ D , find m ∈M so that
F [m] ' d
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Least squares inversion (“the usual suspect”):
given d ∈ D, find m ∈M to minimize
JLS [m] = ‖F [m]− d‖2[+ regularizing terms]
(‖ · ‖2 = mean square)
[Jackson 1972, Bamberger et al 1979, Tarantola &Vallette 1982,...]
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Known since 80’s:
I tendency to get trapped in “local mins”
I transmission modeling more linear thanreflection modeling, so JLS more quadratic
I continuation [low frequency → high frequency]helps
[Gauthier et al. 1986, Kolb et al. 1986]
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The critical step is the first:
initial model ⇔ data bandwidth
⇒ tomography, either waveform or travel time
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Agenda
FWI
Extended Modeling
Extended Modeling and WEMVA
Extended Modeling and FWI
Nonlinear WEMVA and LF Control
Summary
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M = physical model space
M = bigger extended model space
F : M → D extended modeling operator
Extension property:
I M⊂ MI m ∈M⇒ F [m] = F [m]
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Acoustics: m = κ/ρ:
∂2t p − κ∇2p = f
F [m] = p sampled at receiver positions r
f depends on source position s
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Extended acoustics via “survey sinking” - m is anoperator,
(m∇2p)(x) =
∫dym(x, y)∇2p(y)
M⊂ M: multiplication by m(x) ∼ application of
m(x, y) = m(x)δ(x− y)
Physical meaning: action at a positive distance
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Agenda
FWI
Extended Modeling
Extended Modeling and WEMVA
Extended Modeling and FWI
Nonlinear WEMVA and LF Control
Summary
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Born approximation about physical (non-extended)background model
m(x, y) = m(x)δ(x− y) + δm(x, y)
then p ' p + δp,
∂2t δp −m∇2δp =
∫dyδm(x, y)∇2p(y, t)
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Born modeling (derivative of F) = samples of δp atreceiver locations r
G (x, y, t) = Green’s function
δp(s, r, t) =
∫dτ
∫dxG (r, x, t − τ)
×∫
dyδm(x, y)∇2p(s, y, τ)
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=
∫dx
∫dy
[∫dτG (r, x, t − τ)∇2p(s, y, τ)
]×δm(x, y)
⇒ adjoint of Born modeling = imaging operatorapplied to data residual δd(s, r, t)
I (x, y) =
∫ds
∫dr
∫dtδd(s, r, t)
×[∫
dτG (r, x, t − τ)∇2p(s, y, τ)
]
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Receiver wavefield (back-propagate receiver traces)
R(s, x, τ) =
∫dt
∫drδd(s, r, t)G (r, x, t − τ)
Source wavefield
S(s, y, τ) = ∇2p(s, y, τ)
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I (x, y) =
∫ds
∫dτR(s, x, τ)S(s, y, τ)
I propagate receiver field to sunken receiverposition x
I propagate source field to sunken sourceposition y
I cross-correlate at zero time lag
I sum over sources
[Claerbout 1985]
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I Image formation possible for any backgroundmodel m, data residual δd : I = I [m, δd ]
I image I [m, δd ] is actually a model update
I updated model m + αI is physical if it isconcentrated on diagonal x = y (zero offset)
Conclusion: background model m consistent withdata residual δd
⇔ image I [m, δd ](x, y) focused on zero offset locusx = y
⇒ WEMVA
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WEMVA via optimization:
I choose a function φ on M so that (i) φ ≥ 0,(ii) φ[m] = 0⇔ m ∈M
I minimize φ(I [m, δd ]) over m
Typical choice: choose operator A on extendedmodel space M so that M = null space of A,φ[m] = ‖A[m]‖2
[de Hoop and Stolk 2001, Shen et al. 2003, 2005,Albertin et al 2006,...]
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edging towards inversion...
recall that I [m, δd ] updates δm - in fact is a Borninversion, with care!
So paraphrase inversion as:
amongst extended models that fit data residual(under Born modeling), find physical one (perhapsby minimizing φ)
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Agenda
FWI
Extended Modeling
Extended Modeling and WEMVA
Extended Modeling and FWI
Nonlinear WEMVA and LF Control
Summary
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Extended FWI problem:
given d , find m ∈ M so F [m] ' d
I extended FWI is too easy - many solutions!
I extension property: a physical model is anextended model
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Inversion paraphrase:
amongst extended models which fit data, find aphysical one
sounds just like WEMVA!
Difference: full wave field modeling/inversion,rather than Born
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WEMVA-like, using φ:
minimize φ[m] subject to ‖F [m]− d‖ ' 0
Contrast with FWI:
minimize ‖F [m]− d‖ subject to φ[m] ' 0
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All in the family:
Jσ[m, d ] =1
σ‖F [m]− d‖2 + σφ[m]
σ →∞ ⇒ FWI
σ → 0 ⇒ “nonlinear WEMVA”
Gockenbach 1995: path of minima m[σ] can befollowed from small σ to large, leads to FWIsolution provided that small-σ problem can besolved 〈fine print〉
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Agenda
FWI
Extended Modeling
Extended Modeling and WEMVA
Extended Modeling and FWI
Nonlinear WEMVA and LF Control
Summary
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Natural strategy: optimize limσ→0 Jσ using agradient method
BUT for small σ, gradient points mostly indata-consistency direction - can lead to inefficientsolves
Better: compute updates within the data-consistentextended models
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IF VLF band [0,?] Hz were available, acoustic,elastic extended inversion always solvable - noambiguity, can always start frequency continuation
But VLF typically not recorded - what to do?
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IF VLF band [0,?] Hz were available, acoustic,elastic extended inversion always solvable - noambiguity, can always start frequency continuation
But VLF typically not recorded - what to do?
Our solution: make them up!
[non-observed VLF parametrize data-consistentextended models]
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I choose low frequency control model mLF ∈MI choose low frequency source complementary to
data passband, build low frequency modelingoperator FLF , also extended FLF
I create low frequency syntheticsdLF = FLF [mLF ]
I replace d with full bandwidth datadfull = d + dLF
I build full bandwidth modeling operatorFfull = F + FLF
I solve full bandwidth extended inversionFfull[m] ' dfull
I solution m depends on mLF and d
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LF control model parametrizes data-consistentextended models
[similar: migration macro-model parametrizesextended images]
Nonlinear WEMVA objective:
JNMVA[mLF , d ] = φ[m[mLF , d ]]
equivalent to limσ→0 Jσ
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Examples: D. Sun PhD thesis (2012)
Extended modeling by data gathers: model eachgather independently
WE solver, economy ⇒ source gathers (s)
Extended model space M for acoustics = {m(x, s)}
Physical model = independent of s; natural choiceof A = ∇s
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Three layer bulk modulus model. Top surfacepressure free, other boundaries absorbing
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0
0.5
1.0time
(s)
-0.09 -0.08 -0.06 -0.04 -0.02 0.00 0.02 0.04 0.06 0.08sign(p)*p^2 (s^2/km^2)
-1000 -500 0 500 1000Plane wave data - source param s = slowness - note
free surface multiples
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Prestack RTM = extended model gradient athomogeneous initial model
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Extended model inversion at homogenous initialmodel - φ is mean-square of slowness derivative
[also: look ma no multiples]
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LF control model - 3 steps of LBFGS applied toJNMVA
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Extended model inversion from LF model data,step 3
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Data residual, slowness = 0 panel: left, target data;middle, resimulated data from extended model
inversion; right, residual (11.4% RMS)
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WEMVA → FWI by continuation: σ = 0 to σ =∞in one step!
Use LS fit of (physical model → extended model) toproduce optimal initial model for LS inversion
Follow by standard FWI
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Initial model for FWI, obtained as best least-squaresfit to NMVA extended model inversion
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FWI from NMVA-derived initial model- 60 iterationsof LBFGS, 3 frequency bands, 14% RMS residual
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FWI from homogeneous initial model - 60 iterationsof LBFGS, 3 frequency bands, 27% RMS residual
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Incidental observation - multiple scrubbing effect
Even with incorrect LF control model (e.g.homogeneous), extended model inversion appears tosuppress multiple energy
Not limited to layered models...
Theoretical explanation?
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Laterally heterogeneous model with dome structure
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Gather at x = 1.5 km from pre stack RTM =extended model gradient, homogeneous background
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Gather at x = 1.5 km from extended modelinversion, homogeneous background
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Agenda
FWI
Extended Modeling
Extended Modeling and WEMVA
Extended Modeling and FWI
Nonlinear WEMVA and LF Control
Summary
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Summary
I via extended modeling, see nonlinear variant ofWEMVA, FWI as end members of Jσ family
I getting started: LF control model parametrizesdata-consistent extended models, analogue ofmigration macro-model
I Examples suggest multiple suppression propertyof extended inversion
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Thanks to...
I Dong Sun
I TRIP software developers (IWAVE: IgorTerentyev, Tetyana Vdovina, Xin Wang; RVL:Shannon Scott, Tony Padula, Hala Dajani;IWAVE++: Dong Sun, Marco Enriquez)
I Christiaan Stolk, Maarten de Hoop, BiondoBiondi, Felix Herrmann, Tristan van Leeuwen,Wim Mulder, Herve Chauris, Peng Shen, UweAlbertin, Rene-Edouard Plessix,...
I Sponsors of The Rice Inversion Project
I National Science Foundation (DMS 0620821,0714193)