Model Parameterization in tomography problems. Lecture 4

Июль 26, 2022

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Model Parameterization in tomography problems. Lecture 4

Содержание

2. Course structure (major elements): Linearization, reducing to a system of linear equations Inversion BASIC_TOMO code: linear
3. Lecture 4 Model Parameterization in tomography problems Ivan Koulakov IPGG
4. Discretization (parameterization) Presenting the slowness distribution with a finite number of parameters (e.g., cells) System of
5. System of linear equations in tomography General representation: or A is the sensitivity matrix Properties of
6. Common myth: Number of parameters should be less than the number of data Dias N.A., et
7. Synthetic model Effect of grid spacing 2646 rays, all directions Smoothing: 10 Noise: 0 Grid: 1x1
8. Parameterization is a method for definition of a model with a finite number of parameters The
9. Parameterization of the 1D model Layers with constant velocities:
10. velocity in nodes; lineal interpolation in between Parameterization of the 1D model
11. Coefficients of the Taylor series (polynomes) Parameterization of the 1D model
12. Parameterization of the 1D model
13. Parameterization of the 1D model
14. Parameterization of the 1D model
15. 3D model parameterization Three-linear interpolation x1
16. Parameterization Nodes of the parameterization grid are defined according to the ray distribution Between the nodes,
17. Model of Paleozoic complex created by the PetroTrace Company (example of a flexible manipulation with parameterization)
18. Travel times: observed (black) and calculated in the final velocity model (red). Rays in the starting
19. The target object: velocity distribution in the Paleozoic basement It is presumed that the interface geometry
20. Parameterization in zones: Removal links between zones makes it possible to reveal sharp contrasts in velocity
21. Inversion results with apriori known distributions of velocities in zones 1 and 2 1 2 3
22. Inversion results with apriori known distributions of velocities in zones 1 and 2 1 2 3
23. Results of inversions based on different starting models taking into account various apriori information no apriori
24. Regional model parameterization Cells in the Cartesian coordinates Performing inversions in several grids with different basic
25. Regional model parameterization Nodes in the Cartesian coordinates Performing inversions in several grids with different basic
26. Adapting parameterization to the existing model information: Double-sided nodes on interfaces.
27. Parameterization of the global model
28. Irregular grids (Bijwaard, Spakman et al., 1998)
29. Irregular grid gives a balanced solution depending on the ray coverage (no data – no result)
30. Parameterization of the global model
31. Parameterization of the global model
32. Parameterization of the global model
33. Fourier series decomposition: Problem: extrapolation to areas where there are no data!
34. Parameterization with spherical functions Сферические функции: Расчет матрицы первых производных:
35. Parameterization with spherical functions
36. Dziewonski, 1984 Parameterization with spherical functions
38. Скачать презентацию

Слайд 2

Course structure (major elements):
Linearization, reducing to a system of linear equations
Inversion
BASIC_TOMO

code: linear rays, inversion

Adaptive parameterization

Ray tracing

PROFIT code: iterative active source tomography

Source locations

LOTOS code: iterative passive source tomography

Ambient noise

Surface waves

Surface wave tomography algorithms

Practical exercises:

Theoretical blocks:

Слайд 3

Lecture 4 Model Parameterization in tomography problems
Ivan Koulakov
IPGG

Слайд 4

Discretization (parameterization)

Presenting the slowness distribution with a finite number of parameters

(e.g., cells)

System of linear equations:

Слайд 5

System of linear equations in tomography
General representation:

or
A is the sensitivity matrix
Properties

of the tomography system (classical theory):

The system is overdetermined: M>N (not valid in practice);
The system is underdetermined: rank(A)The data vector contains noise.

Solution using Least Square Methods

Slowness:

Слайд 6

Common myth: Number of parameters should be less than the number

of data

Dias N.A., et al., (AZORES), Tectonophysics, 445, 301–317

S. Husen et al. (Yellowstone Park) Journal of Volcanology and Geothermal Research (2004) 397-410

In this case, the results are strongly grid-dependent. Shifting the grid leads to completely different results.

Слайд 7

Synthetic model
Effect of grid spacing
2646 rays, all directions
Smoothing: 10
Noise: 0
Grid: 1x1
Grid:

10x10

Grid: 5x5

The parameterization should have the minimum effect on the result. The grid spacing should be much smaller than the minimum resolved structures

Слайд 8

Parameterization is a method for definition of a model with a

finite number of parameters

The function is presented as a decomposition with a set of basis functions fi(r) and coefficients ci.
In a case of cell parameterization, fi(r)=1 inside the i-th cell and fi(r)=0 outside the cell.
There might be many types of the basis functions.

Слайд 9

Parameterization of the 1D model
Layers with constant velocities:

Слайд 10

velocity in nodes; lineal interpolation in between
Parameterization of the 1D model

Слайд 11

Coefficients of the Taylor series (polynomes)
Parameterization of the 1D model

Слайд 12

Parameterization of the 1D model

Слайд 13

Parameterization of the 1D model

Слайд 14

Parameterization of the 1D model

Слайд 15

3D model parameterization
Three-linear interpolation
x1

Слайд 16

Parameterization
Nodes of the parameterization grid are defined according to the ray

distribution
Between the nodes, velocity is approximated using the bi-linear interpolation

V11

V12

V21

V22

y11

y12

y21

y22

x0,y0,V0

V1= V11+ ((V12-V11) / (y12-y11)) * (y0-y11)

V2= V21+ ((V22-V21) / (y22-y21)) * (y0-y21)

V0= V1+ ((V2-V1) / (x2-x1)) * (x0-x1)

Слайд 17

Model of Paleozoic complex created by the PetroTrace Company
(example of a

flexible manipulation with parameterization)

Reconstruction result with the regular PROFIT code for 2D active-source tomography

The result is not perfect. We can recover an anomaly at 22-24 km. The anomaly to the left (12-14 km) and to the right (26-30) are almost not resolvable.

Starting 1D velocity model

Слайд 18

Travel times: observed (black) and calculated in the final velocity model

(red).

Rays in the starting 1D model and
nodes of the parameterization grid

Rays in the final velocity model after the 9th iteration

Слайд 19

The target object: velocity distribution in the Paleozoic basement
It is presumed

that the interface geometry and velocity distributions in the upper part can be robustly retrieved by reflection seismics.

Слайд 20

Parameterization in zones:
Removal links between zones makes it possible to reveal

sharp contrasts in velocity anomalies.
The nodes in different zones can be attributed with different weights (e.g., in the upper part, the retrieved anomalies are much weaker).

Слайд 21

Inversion results with apriori known distributions of velocities in zones 1

and 2

Слайд 22

Inversion results with apriori known distributions of velocities in zones 1

and 2

Слайд 23

Results of inversions based on different starting models taking into account

various apriori information

no apriori information

true model

known structures in zone 1

known structures in zones 1 and 2

Слайд 24

Regional model parameterization
Cells in the Cartesian coordinates
Performing inversions in several grids

with different basic orientations
Then create an average model, which is almost not affected by grid geometry

Слайд 25

Regional model parameterization
Nodes in the Cartesian coordinates
Performing inversions in several grids

with different basic orientations
Then create an average model, which is almost not affected by grid geometry

Слайд 26

Adapting parameterization to the existing model information:
Double-sided nodes on interfaces.

Слайд 27

Parameterization of the global model

Слайд 28

Irregular grids (Bijwaard, Spakman et al., 1998)

Слайд 29

Irregular grid gives a balanced solution depending on the ray coverage

(no data – no result)

Irregular grids (Bijwaard, Spakman et al., 1998)

Parameterization of the global model

Слайд 30

Parameterization of the global model

Слайд 31

Parameterization of the global model

Слайд 32

Parameterization of the global model

Слайд 33

Fourier series decomposition:
Problem: extrapolation to areas where there are no data!

Слайд 34

Parameterization with spherical functions
Сферические функции:
Расчет матрицы первых производных:

Слайд 35

Parameterization with spherical functions

Слайд 36

Model Parameterization in tomography problems. Lecture 4

Содержание

Course structure (major elements):Linearization, reducing to a system of linear equationsInversionBASIC_TOMO

Lecture 4 Model Parameterization in tomography problems Ivan KoulakovIPGG

Discretization (parameterization) Presenting the slowness distribution with a finite number of parameters

System of linear equations in tomographyGeneral representation: orA is the sensitivity matrixProperties

Common myth: Number of parameters should be less than the number

Synthetic modelEffect of grid spacing2646 rays, all directionsSmoothing: 10Noise: 0Grid: 1x1Grid:

Parameterization is a method for definition of a model with a

Parameterization of the 1D modelLayers with constant velocities:

velocity in nodes; lineal interpolation in betweenParameterization of the 1D model

Coefficients of the Taylor series (polynomes)Parameterization of the 1D model

Parameterization of the 1D model

Parameterization of the 1D model

Parameterization of the 1D model

3D model parameterizationThree-linear interpolationx1

ParameterizationNodes of the parameterization grid are defined according to the ray

Model of Paleozoic complex created by the PetroTrace Company(example of a

Travel times: observed (black) and calculated in the final velocity model

The target object: velocity distribution in the Paleozoic basementIt is presumed

Parameterization in zones:Removal links between zones makes it possible to reveal

Inversion results with apriori known distributions of velocities in zones 1

Inversion results with apriori known distributions of velocities in zones 1

Results of inversions based on different starting models taking into account

Regional model parameterizationCells in the Cartesian coordinatesPerforming inversions in several grids

Regional model parameterizationNodes in the Cartesian coordinatesPerforming inversions in several grids

Adapting parameterization to the existing model information:Double-sided nodes on interfaces.

Parameterization of the global model

Irregular grids (Bijwaard, Spakman et al., 1998)

Irregular grid gives a balanced solution depending on the ray coverage

Parameterization of the global model

Parameterization of the global model

Parameterization of the global model

Fourier series decomposition:Problem: extrapolation to areas where there are no data!

Parameterization with spherical functions Сферические функции:Расчет матрицы первых производных:

Parameterization with spherical functions

Dziewonski, 1984Parameterization with spherical functions

Похожие презентации

Course structure (major elements):
Linearization, reducing to a system of linear equations
Inversion
BASIC_TOMO

Lecture 4 Model Parameterization in tomography problems
Ivan Koulakov
IPGG

Discretization (parameterization)

Presenting the slowness distribution with a finite number of parameters

System of linear equations in tomography
General representation:

or
A is the sensitivity matrix
Properties

Synthetic model
Effect of grid spacing
2646 rays, all directions
Smoothing: 10
Noise: 0
Grid: 1x1
Grid:

Parameterization of the 1D model
Layers with constant velocities:

velocity in nodes; lineal interpolation in between
Parameterization of the 1D model

Coefficients of the Taylor series (polynomes)
Parameterization of the 1D model

3D model parameterization
Three-linear interpolation
x1

Parameterization
Nodes of the parameterization grid are defined according to the ray

Model of Paleozoic complex created by the PetroTrace Company
(example of a

The target object: velocity distribution in the Paleozoic basement
It is presumed

Parameterization in zones:
Removal links between zones makes it possible to reveal

Regional model parameterization
Cells in the Cartesian coordinates
Performing inversions in several grids

Regional model parameterization
Nodes in the Cartesian coordinates
Performing inversions in several grids

Adapting parameterization to the existing model information:
Double-sided nodes on interfaces.

Fourier series decomposition:
Problem: extrapolation to areas where there are no data!

Parameterization with spherical functions
Сферические функции:
Расчет матрицы первых производных:

Dziewonski, 1984
Parameterization with spherical functions