Numerical Meshes from Seismic Images

Karl Apaza AgüeroPaulo Roma Cavalcanti

Antonio OliveiraClaudio Esperança

COPPE – Sistemas - UFRJ

Goal

• Creation of numerical meshes from seismic images.

• Integrates several techniques:

• Image Processing.

• Physical Modeling.

• Optimization.

• Computational Geometry.

Seismic Methods

3

• Based on the emission of acoustic waves onto the surface of the earth or the sea.

• Reflection Method:●Acquisition. ●Processing.● Interpretation.

Traditional Approach

Seismic Geometric Model Mesh

4

Geometric Model = Set of curves and surfaces

● Horizons: separating surfaces between geological layers.

● Faults: discontinuities produced by sliding of layers.

Geometric Model

Alternative Approach

IDEA: Seismic Mesh

5

Generates meshes directly from seismic images.

Avoids the creation of an intermediary geometric model.

Extracts horizons and faults directly from the mesh. Mesh

Method

Enhance the important features.● Image processing techniques.

Volumetric Visualization

Seismic Image

Seismic Data

Enhancement of the Important Features

Initial lattice Generation

Minimization of the Potential Energy

Atom Connection

Aligned Mesh Simulation

Generate an initial lattice of atoms based on the important features.

● Interaction force between atoms.● Pseudo regular lattice.

Minimize the total potential energy function.

● Interaction force between atoms.● Steepest Descent Method.

Connect atoms. ● Delaunay Triangulation / Voronoi

Tessellation.

Atoms

An atom is an image point subjected to forces exerted by its neighbors.

Influence zone depends on a threshold distance D.

An inter-atomic force must satisfy:

7

Interaction among atoms

Properties

Be null beyond a certain distance, limiting the influence zone of an atom.

Be a continuous function of the inter-atomic force.

Be repulsive (positive) to avoid atoms very close to each other.

Be attractive (negative) to avoid large empty spaces, when atoms are far away from each other.

Nominal Distance

Nominal distance, d, is the distance where attraction forces turn into repulsion forces.

Force Model

Interaction force among atoms is a piecewise polynomial function:

d

xxu

ji

10

d: nominal distance.

, normalized distance.

u

uuuuf5.10

5.104

5

8

19

8

9 32

Scalar Potential

To employ minimization techniques:• force is defined as the negative of the

gradient of an scalar potential field.

duufu )()(

u

uuuuu5.10

5.1016

5

24

19

8

9

256

153 43

Atomic Potential Energy

Is the weighted sum of each atom energy in the system.

The atom energy is the sum of the forces exerted onto it by its neighbors:

12

n

i

n

ijj j

ji

n xd

xxxxxAA

1 ,121 2

1,...,,

Image Potential Energy

Is the sum of the potential field of the image pixels associated to atoms.

The potential field of a point, b(xi), depends on

the pixel value (grey level) associated to the image.

n

iin xbxxxBB

121 ,...,,

Total Potential Energy Is the weighted sum of the atomic potential

energy and the image potential energy:

The scale, ß, determines the relative contribution of A and B.

• ß=0 atoms create a regular lattice, not necessarily aligned to the important features.

• ß=1 atoms are sensitive only to the important features, producing a highly irregular lattice.

Depends on the type of atom connection.14

BAxxxPP 1,...,, 321

Enhancement of the features

Sobel Detector Identifies important

features on the image.

• Image differentiation.• Image smoothing.

15

Morphological Operators Dilation and erosion

operators enhance the important features.

Thicken or thin important features.

Erosion: 3 x 3 mask

Initial lattice

16

The initial lattice of atoms should have the following characteristics:

• Minimize, locally, the atomic potential energy.

• Be highly regular.• Be consistent with the nominal distance

function.

Nominal Distance Function

For a constant function, it is easy to obtain a regular initial lattice holding the previous properties.

A rectangular lattice is the simplest choice. An hexagonal lattice is the best solution for an initial

lattice of points. A non-constant function poses some difficulties.

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Seismic Image

Nominal distance function dmin = 6 (black pixels)

dmax = 12 (white pixels)

Make an array of boolean flags, w(x)=false

Create an empty list of atoms Create an empty queue of atom positions Add to the queue the position of the

image centre While the queue is not empty

• Get, and remove from the queue, the first position xi

• If xi is onto the image limits

• Make an sphere with centre xi and diameter d(xi)

• If the sphere contains positions with w(x)=false

• Do for all positions inside the sphere w(x)=true;

• Add to the list an atom with coordinates xi;

• Add at the end of the queue ideal positions for neighbors

Algorithm: Pseudo-

regular lattice of

atoms

Minimization of The Energy Function

After the creation of the initial lattice, the atoms must be moved to a configuration that minimizes the total potential energy, P.

The Steepest Descent Algorithm (SDA) is used to minimize the total potential energy function, which may possess several local minima.

The search is repeated until the best minimum is found.

Lattice Optimizer

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The threshold Є controls the iterations until the decreasing in P is negligible.

Seismic Image

Disturbance = 0.2 x d

Get the initial lattice x1, x2, ..., xn

Compute the total potential energy of the initial lattice, P

Do {

• P0 = P

• Disturb x1, x2, ..., xn

• Do {

• Pi = P

• One step of the SDA algorithm

} While Pi – P > Є |Pi|

} While P0 – P > Є |P0|

Delaunay Triangulation The optimized lattice of

atoms is structured through a Delaunay triangulation or a Voronoi tessellation.

Both schemes tend to create edges (in 2D) and faces (in 3D) aligned to the important features of the image.

A Delaunay triangulation always connects atoms to its closest neighbors. 21

Delaunay Triangulation: 545 atoms

Voronoi Tessellation

22

Voronoi connects circumcentres of Delaunay triangles.

Atoms concentrate near the boundaries of the important features.

23

Voronoi

Tessellation

652 atoms

Initial latticedmin= 5

dmax = 10

Optimized

lattice

Dist. = 0.2 x d

Voronoi

Tessellation

onto

optimized

lattice.

Results

24

Delaunay Triangulation

495 atomsdmin = 10 (black pixels)dmax = 20 (white pixels)

Disturbance = 0.1 x d

Voronoi Tessellation

775 atomsdmin = 8 (black pixels)

dmax = 16 (white pixels)Disturbance = 0.1 x d

Results

25

Sobel

3 x 3

Dilation

3 x 3Brain

Results

26

Final Meshpixel color = triangle circumcentre

Optimized latticedmin = 3 (black pixels)dmax = 9 (white pixels)Disturbance = 0.2 x d

Mesh1700 atoms

Results

27

Delaunay Triangulation generated

for the seismic volume of The

Stratton Field, South of Texas.

Conclusions

28

The enhancement of the important features is fundamental to the presented method, in order that the point optimizer produce good results.

For an image with smooth luminance, the method is able to align the mesh to the important features.

Conclusions

29

The presented parameters can be applied to a great number of images.

If the input image do not allow the closing of regions, maybe because it was not filtered appropriately, the method does not close the “holes".

The method can be used to segment a large range of images.

Main References

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[Hale2001] “Atomic images – A Method for Meshing Digital Images”. Proceedings of the 10th International Meshing Roundtable, pp. 185-196. 2001.

[Hale2002] “Atomic meshes: from seismic imaging to reservoir simulation”. Proceedings of the 8th European Conference on the Mathematics of Oil Recovery. 2002.

[Jalba2004] “CPM: A Deformable Model for Shape Recovery and Segmentation Based on Charged Particles”. IEEE Transactions on Pattern Analysis and Machine Intelligence. 2004.