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The Art of Inverse Probleminferring model parameters from output data

- Inverse problems are among the most challenging in computational and applied science and have been studied extensively.
- Although there is no precise definition inverse problems are concerned with the determination of inputs or sources from observed outputs or responses.
- This is in contrast to direct problems, in which outputs or responses are determined using knowledge of the inputs or sources.

2

Presentation outline

“Gauss-Newton and full Newton methods in frequency-space seismic waveform inversion”

Pratt, Shin, Hicks

Geohpys.J.Int. (1998) 133, 341-362

“High resolution velocity model estimation from refraction and reflection data”

Forgues, Scala, Pratt

SEG 1998

“Seismic Waveform inversion in the frequency domain”

Pratt, Geophysics Jan 11, 1999

“Nonmonotone Spectral Projected Gradient Methods in Convex Set”

1999, Birgin, Martinez, Raydan

- inversion framework
- mathematical framework
- steepest descent optim.
- lagrangian approach
- optimization loop
- newton optimization
- conjugate direction opt.
- 1d optimization
- constrained optimization
- test cases

“Multiscale seismic waveform inversion”

Bunks, Saleck, Zaleski, Chavent

Geohpysics (1995) 60, 1457-1473

“Nonlinear inversion of seismic reflection data in a laterally invariant medium”

Pica, Diet, Tarantola,

Geohpysics (1990) 55, 284-292

“Pre-stack inversion of a 1D medium”

Kolb, Collino, Lailly

IEEE (1986) 74, 498-508

3

Inversion framework

- Parameters: NxNyNz unknowns to recover: the velocity field c(x,y,z)
- Observed data/measurements: recorded data at a reference depth STACK(x,y,t) = P(x,y,z=0,t).
- Simulated data: wave-field propagation imposed by the acoustic wave equation using some trial velocity field
- Inversion: find the velocity field that minimizes some measure of the misfit between observed and simulated data

We solved the inverse problem with a single shot acquisition.

The generalization to multiple shots is straightforward and can result in a

better inversion.

4

Mathematical framework

- measureSTACK(x,y,t) at same reference level z=0, produced by a single source
- try a guessc(0)(x,y,z) for the velocity field
- solving the acoustic wave equation, simulate the pressure field over the entire spatial domain (with adequate B.C. and I.C.)

- evaluate the error or cost function and if necessary its derivatives (cumbersome)

- update iteratively the velocity field, with the intent to minimize the error function

- iterate this procedure up to a fixed “error threshold”

5

Steepest descent optimization

- The velocity updating technique is usually based
- on local informations, e.g. the gradient:

- Fixed point = minima

- problem: avoid local minima

6

Lagrangian approach

- Constrained minimization problem adjoint field

A sort of wave

equation with source

term = residual error

Wave

equation

Back in time!

From P and l evaluate the gradient

PROBLEM: time alignment!

7

BMod

AdjMod

Optimization loopdo it=0, nit-1

!

call FMod

do step = 0, nt-1

!

call BMod

call LoadMeasField

call AdjMod

call PartialGrad

call PartialCostF

!

end do

call Optimizer

!

end do

Record data at z=0 & on the boundaries

Use information on the boundaries to backpropagate field P

Load observed data

With real and simulated measurements build the source term and solve for the adjoint field l

Evaluate partial cost function and gradient

Update velocity field

Inner loop: align in time both direct and adjoint fields to perform in-core gradient evaluation

8

Newton optimization

- The optimization procedure can use also information from the Hessian (second derivative matrix) but this is very expensive for both computational (# direct propagation = # parameters) and storage requirements ( [NxNyNz]2 )!
- E.G. Newton, Quasi-Newton or Gauss-Newton methods:
- Thus, aiming to a 3D reconstruction, we decided to only use the gradient.

9

Conjugate direction optimization

- To achieve better convergence we studied different conjugate direction algorithms:
- [1] Fletcher-Reeves
- [2] Polak-Ribiere
- [3] Hestenes-Stiefel
- (but we have not observed sensible differences)

[1]

[2]

[3]

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1D optimization

- At each iteration step, for each fixed direction d and velocity c, find a scalar asuch that the resulting error function (depending now on a single real parameter)
- be minimum,
- E.G. by line search, bisection, generalized decreasing conditions

12

Constrained optimization

- Because of the box constraints over the velocity,
- we are forced to adopt the projected conjugate gradient:

13

nx = 116

nz = 66

nt = 270

dx = 3.

dz = 3.

dt = 0.00065

thick_x = 0

thick_z = 0

rec_thick_x = 1

rec_thick_z = 1

z_record = 4

Nopt = 20

Niter = 100

We will consider inversion of small 2D synthetic

data-sets.

For a better tuning of the algorithms we used

velocity field with no lateral variations, but the

code is genuine 2D.

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constant function

Very good result,

small changes after

140 iterations ...

Initial guess: straight line

15

The cost function decreases

of about 4 orders of magnitude.

The steepest slope is obtained

in the first ~20 iterations.

A second sudden jump comes as

the velocity gets the second ridge!

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constant function

After ~10 iterations

we get the first ridge ...

Initial guess: straight line

17

We take one of the last iterated

field (#11) and freeze the gradient

of the first (20) layers

In ~20 iterations we reach both the first and the second ridges!

19

Target: piecewise

constant function

Things goes wrong if the

low frequency trend is not

included in the

initial guess ...

Initial guess: straight line

but it does not

matches the ‘trend’

21

from #2 of previous

iterations ...

Freezing the first 20 layers, the

1st discontinuity gets worse but

we better recover the 2nd one ...

23

Target: parabola

Good!

After ~170 iterations

things does not

change too much!

25

we get a really

good result!

Target: parabola

We start from the previous

velocity field (#60) and freeze

the gradient at the first layers

(#20)

27

Initial guess: straight line

Target:

parabola + sin

Nice!

The greatest part is done

in the first ~100 iterations!

29

Not good as before: we only get

the medium trend!

Target:

parabola + sin

We start from a previous iteration

(#20) and freeze the gradient at

the first (20) layers

31

- The main problem is the presence of a large number of local minima. To get rid of them is possible to linearize the direct model (eg Born approx.), to have a convex cost function
- or adopt some multi-scale approach:
- large to small spatial scale, or
- low to high time frequencies

but: loose refracted/multiply reflected waves, ecc. ecc.

but: the ultra-low frequency (the velocity field trend) components don’t produce reflected waves, thus must be already present in the initial guess.

33

Time versus Frequency Domain

The advantage of the time domain is the intuitive comprehension of the involved fields and results

- Advantages of the time frequency domain
- high data compression rate (~10)
- uncoupled problems in embarassing parallelism
- large to small spatial scale approach, inverting separately small and large frequenciesquickest and scalable approach

34

Spectral conjugate gradient

- Spectral Conjugate Gradient Method
- the advantage is that in this way the conjugate direction (- g) contains some explicit information on the Hessian matrix.

36

Modified Nonlinear Conjugate Gradient

- In geophysic application, the number of parameters is very large, this motivates the choice of a conjugate gradient minimization algorithm
- Without uphill movements (a<0) in the line search procedure, none optimization method will prevent the trapping inside the local minima

37

- In our approach a can be either positive, describing a movement in the descent direction of pk, or negative.
- For a negative, the line search is similar

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- very noisy function, presenting oscillations up to small scales (many local minima)
- after 7 steps both Wolfe conditions are satisfied

Allowing a<0, the algorithm can visit and leave most local minima

39

- the function is a sum of a simple convex quadratic and low-amplitude high frequency perturbation (N=2)
- after 8 steps both Wolfe conditions are satisfied

Allowing a<0, the algorithm can visit and leave most local minima

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- same function as before, with N=32
- standard gradient based minimization methods are not satisfactory with such a noisy function
- on nontrivial analytical examples, our approach converges quickly towards the global minimum

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The number of parameters plays a crucial role in the choice of the algorithm to minimize the cost function j(p) in the parameter space

Without uphill movements in the line search procedure, none optimization method will prevent the trapping inside the local minima

The landscape of the cost function presents many local minima

storage

convergence

42

- The number p of parametersimpacts on the choice of the optimization strategy:
- for very small p the gradient can be computednumerically
- for small p, use the gradient and the Hessian to compute the search directions
- exact Hessian (Newton)
- approximation of the Hessian as the iteration progresses (Quasi-Newton).
- for large p, use only the gradient to compute the search directions
- nonlinear conjugate gradient
- for very large p, use stochastic methods
- simulated annealing

43

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