ipam mga tutorial: multiscale problems in...

IPAM MGA Tutorial: MultiscaleProblems in Geophysics

Naoki [email protected]

http://www.math.ucdavis.edu/˜saito/

Department of Mathematics

University of California, Davis

Sep. 2004 – p.1

Acknowledgment

Nick Bennett (Schlumberger)

Bob Burridge (MIT)

Charlie Flaum (Schlumberger)

T. S. Ramakrishnan (Schlumberger)

Anthony Vassiliou (GeoEnergy)

Sep. 2004 – p.2

Outline

Multiscale Nature of Geophysical Measurements

Importance of Geometry, Textures, and Clustering:High Relevance to MGA2004

Large Scale: Seismic Signal/Noise Separation

Small Scale: Fracture and Vug Detection fromElectrical Conductivity Images

High-Dimensional Clustering Problem in ExplorationGeophysics

Summary

Sep. 2004 – p.3

Multiscale Nature of Geophysical

Measurements

Large Scale: Surface Seismic Survey:

used for structural mapping of the entire reservoirvolume

wide spatial coverage: � 10km x 10km x 3km

spatial sampling: 25m x 25m

can resolve features > 30-40ft (9-12m)

vertical axis: time

Sep. 2004 – p.4

Surface Seismic Survey

Sep. 2004 – p.5


Measurements . . .

Medium Scale: Borehole Seismics:

receivers are in a borehole/well

can record not only reflected waves but also directwaves

can record reflected waves with less attenuation

used for linking surface seismic information and welllog information (e.g., sonic logs)

depth sampling: 20-50m (wish: < 10m)

Sep. 2004 – p.6

Borehole Seismics

courtesy: Schlumberger Sep. 2004 – p.7

Borehole Seismics . . .

courtesy: Schlumberger

Sep. 2004 – p.8


Measurements . . .

Small Scale: Wireline measurements, cores:

used for detailed geological/geophysical informationin the vicinity of wells

spatial coverage: � 1m x 1m x 3km

can resolve features � 0.5-3ft (0.15-1m) or less

vertical axis: depth

cores provide absolute truth, but expensive

can characterize not only elastic but also electricaland nuclear properties of subsurface formations

Sep. 2004 – p.9

Electrical Conductivity Image ( � 2.5ftx 2.3ft)


Electrical Conductivity Imaging Tool

��

��

FMI measurement current path.

(a) Tool (b) Pad/Flap



Measurements . . .

Large Scale Geometry

Medium/Small Scale Geometry

Textures (Stratigraphy, Lithology, Facies)

Reconciliation of multiscale info for better models

Highly Relevant to MGA 2004

This tutorial: introduction of problems + some attemptswith older techniques

My hope: provoke the audience so that some of youmay be interested in applying new techniquesdiscussed in MGA 2004 to these problems!

Sep. 2004 – p.12

Extraction of Large Scale Geometric Objects

Harlan, Claerbout, & Rocca (1984)

Motivation: Separate geologic events (layerboundaries) from other events and noise. Other eventscould be interference patterns, scattering from faults inthe form of hyperbolas, etc.

Measurement: Seismic signals (large scale)

Beginning of “multilayered transforms,” “coherentstructure extraction,” and ICA

Method: iteration of “focusing” transforms + softthresholding

Sep. 2004 – p.13

Harlan, Claerbout, & Rocca (1984) . . .

Define “focusing” as an invertible linear transformationmaking the data more statistically independent (or lessstatistically dependent)

A transform focusing signal (pattern) must alsodefocus noise

Focused signal becomes more non-Gaussian;defocused noise becomes more Gaussian

In the transformed domain, apply filtering orsoft-thresholding operations to extract signal orseparate signal from noise

Sep. 2004 – p.14

Harlan, Claerbout, & Rocca (1984) . . .

A stacked seismic section = of

Geologic component � linear events

Diffraction component � hyperbolic events

Noise component � white Gaussian noise �� .

Sep. 2004 – p.15

Radon Transform

The 2D-line version for

� �� can be stated as follows:

"! �$#� % � & � �� ' �( � '*) # ) %� �+ � + '-,

Has a huge list of applications (X-ray tomography,geophysics, wave propagation, . . . )

Can be generalized: integration of an .-dimensionalfunction over /-dimensional geometric objects

Inversion formulas exist as Beylkin showed yesterday

Lines in an image are transformed to sharp peaks in�$#� % � -domainSep. 2004 – p.16

Example 1

(a) Original (b) 021 3

(c) Randomized (d) 01 3

Sep. 2004 – p.17

Example 1 . . .

(e) Thresholded (f) Recon

(g) Residual (h) Hyp. Ext.Sep. 2004 – p.18

Example 2: Original

Sep. 2004 – p.19

Example 2: Reconstruction

Sep. 2004 – p.20

Example 2: Residual

Sep. 2004 – p.21

Challenges

Fast Radon and Generalized Radon Transforms & 4

Beylkin’s Fast Radon Transforms and USFFT

Apply curvelets/ridgelets & 4 F. Herrmann

3D is a big issue

Sep. 2004 – p.22

Fracture Plane Detection fromBorehole Images

Objective: detect and characterize fracture planesstriking a borehole

Measurement: borehole images (either electric oracoustic), medium 5small scale

Method: Hough transform

Fracture plane can be parameterized by6� � 7 � ��8 & 9,

The name of the game is to estimate the parameters� 6� 7� 9 �from available borehole images.

Sep. 2004 – p.23

Fracture Plane Cutting a Borehole

RO

x

y

z

(o,o,d)φ

α

(xi,yi,zi)x

x

ηi

θi

Sep. 2004 – p.24

A Synthetic Borehole Image(Unfolded)

0 100 200 300

050

100

150

200

250

300

Sep. 2004 – p.25

Detected Edges

0 100 200 300

050

100

150

200

250

300

Sep. 2004 – p.26

An Edge Element vs FractureGeometry

An edge location

��2: � �: � 8 : �

and an edge orientation ;:

constrain the fracture plane:

6�: � 7 �: ��8 : & 9

6 �: ) 7�: & <>= ? ;: ,

These two equations specify a straight line in the� 6� 7� 9 �

-space:

6 & 6 � 9 � & 9) 8 : @ �: � �: <>= ? ;:

7 & 7 � 9 � & 9) 8 : @ �: ) �: <= ? ;: ,

Sep. 2004 – p.27

Hough Transform

is a popular technique to detect certain geometric patternsfrom the edges in images. Informally, it is related to Radontransform. For line detection in 2D:

A � 6� 7 � & B �CD E � �� C �( � 7 � 6� ) � �+ � + ��

where

B

is a thresholding operation to make a binary image,D E is a regularized derivative with a characteristic scale F.But the HT more heavily utilizes the duality between the fea-

ture space and the parameter space.

Sep. 2004 – p.28

Hough Transform . . .

Hough (1962) patented the original method for binaryimages

Ballard (1981) generalized to arbitrary shape withgradient info

Illingworth & Kittler (1988) surveyed and listed 136papers

Leavers (1993) surveyed and listed 173 papers

Sep. 2004 – p.29


Compute edge positions and orientations (gradient)

Prepare the parameter space (called accumulatorarray) whose axes are the parameters specifyingshapes (e.g.,lines: gradient and �-intercept, circles:center and radius)

For each edge, vote for all possible (specific) shapes inthe accumulator array

Accumulate the votes for all (significant) edges

Local maxima in the accumulator array identifyconcrete shapes

Sep. 2004 – p.30


+ Very robust, insensitive to broken patterns andnoises

+ Can detect multiple or intersecting objects

+ Can be more effective given edge orientationinformation

- Computationally slow (voting process)

- High storage (memory) requirement

Discretization of parameter space G templatematching in feature space

Sep. 2004 – p.31

Edge Orientation ConstrainsPossible Planes

0 100 200 300

050

100

150

200

250

300

Sep. 2004 – p.32

Detected Fracture Planes

0 50 100 150

050

100

150

200

250

300

Sep. 2004 – p.33

Real Data Result


Detection of Vugs/Elliptic Shapes

Objective: detect and characterize vugs (ellipticalcavity or void in a rock) & 4 porosity, depositionalinformation

Measurement: borehole images (electric or acoustic),small scale

Method: Hough transform for ellipses

Sep. 2004 – p.35

Hough Transform for Ellipses

5 parameters (center coordinate

�� , lengths ofmajor/minor axes

� 6� 7 �

, and orientation of major axis

H

)are required to specify each ellipse

Voting in the 5 dimensional accumulation array is costly

Found lower dimensional strategy if one needs onlycertain combinations of parameters such as centerpositions and areas of ellipses (N. Bennett, R.Burridge, & NS)

Sep. 2004 – p.36

Hough Transform for Ellipses . . .

Each pair of edges in an image determines aone-parameter family of ellipses (an exercise ofprojective geometry).

Let

IKJ & �� J� � J � and L J & � % J� M J � , N & O� P

be thepositions and the normal vectors of a pair of edges.

Sep. 2004 – p.37


Then, the following represents a conic section

Q �� :

Q �� R S � T & U @ �� ) S VXW �� V @ �� & Y�

where

U �� T &Z[Z[Z\Z[Z[Z[Z[Z� � O

� W �W O

� @ � @ OZ[Z[Z\Z[Z[Z[Z[Z

& Y

is the lineIW I @ ,

V J �� T & % J �� ) � J � � M J � � ) � J � & Y

is the tangent line at

I J . Sep. 2004 – p.38


If

S^] � Y� S: �

, then

Q �� R S �

represents an ellipsewhere

S: & _ � L W ` ) ) aIW I @ � � L @ ` ) ) aI @ IW �b � %^W M @ ) % @ MW � @ ,

Other cases:

Sdc Y

orSde S: : hyperbola;

S & Y

: lineIW I @ ; S & S: : parabola.

All the geometric quantities of interest (e.g., all those 5parameters as well as its area: f 6 7) can be written as afunction of

S.

Sep. 2004 – p.39


0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

10

Sep. 2004 – p.40


Position and area of ellipses are of particular interest

Need to consider only

g h

curve��: � S �� : � S �� f 6 � S � 7 � S � �

instead of resolving 5parameters

For each pair of all the strong edges (or its randomsubset), vote for all ellipses lying along this curve

Pick the local maxima in this 3D voting space(accumulator array)

Sep. 2004 – p.41

Five Ellipses

0.0 0.2 0.4 0.6 0.8 1.0

10.

80.

60.

40.

20

(a) Original

0.0 0.2 0.4 0.6 0.8 1.0

10.

80.

60.

40.

20

(b) Edge Mag.

0.0 0.2 0.4 0.6 0.8 1.0

10.

80.

60.

40.

20

(c) Edge Ori.

0.0 0.2 0.4 0.6 0.8 1.0

10.

80.

60.

40.

20

(d) Detected Cnt. Sep. 2004 – p.42

Five Ellipses: Orientation vs Area

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

0.0

0.2

0.4

0.6

0.8

1.0

ORIENTATION

AR

EA

Sep. 2004 – p.43

Challenges

Fast Hough Transform via multiscale?

Will beamlets or its generalized version help?

Other geometric shapes?

Remember that there are strong needs in manyapplications to specific geometric shapes from images!

Sep. 2004 – p.44

A Library of FocusingTransformations

Each focusing transformation focuses a specificgeometric object

Can repeat extraction of different objects sequentially

A Library of Focusing Transformationsh 6 ' 6 & i 7jlk m 'on W � i 7jlk m 'on @ � ` ` ` ,e.g., A borehole image = fractures + vugs + residuals.

Sep. 2004 – p.45

Reconciliation of Different Scales

Small & 4Large: Geometric information from smallerscale measurements serves as constraints for modelsof seismic imaging and inversion

Large & 4Small: Geometry from surface seismic helpswell-to-well correlation (for avoiding “local minima”)

Easier said than done

Sep. 2004 – p.46

Reconciliation of Different Scales . . .


Clustering of Multiscale GeophysicalInformation

Objective: identify clusters of various attributes ofmeasurements; associate them with the facies (rocktypes); compare them with cores

Measurements include:

The standard well logs (porosity, density, . . . )

Statistics (mean, variance, skewness, kurtosis) ofborehole images (electric) with sliding windows;

Wavelet features of borehole images (electric) withsliding windows

Method: Self-Organizing Map (SOM) algorithm

Sep. 2004 – p.48

Wavelet-Based Texture Features

Decompose available images into local frequencycomponents via wavelet packets

Select the best basis that captures majority of energyof images

Represent images based on the selected basisfunctions

Supply the square of the top

p

(say,

O Y Y

) coefficients toSOM

Sep. 2004 – p.49

FaciesC

lassificationfrom

Wavelet

Features

Depth

04*10^7

-9170 -9150 -9130 -9110 -9090-9170 -9150 -9130 -9110 -9090

Depth

04*10^5

-9170 -9150 -9130 -9110 -9090-9170 -9150 -9130 -9110 -9090

Depth

010^5

-9170 -9150 -9130 -9110 -9090-9170 -9150 -9130 -9110 -9090

Depth

04*10^4

10^5

-9170 -9150 -9130 -9110 -9090-9170 -9150 -9130 -9110 -9090

Sep.2004

–p.50

FaciesC

lassificationfrom

Wavelet

Features...

Depth

02*10^4

5*10^4

-9170 -9150 -9130 -9110 -9090-9170 -9150 -9130 -9110 -9090

Depth

010^4

-9170 -9150 -9130 -9110 -9090-9170 -9150 -9130 -9110 -9090

Depth

04*10^3

10^4

-9170 -9150 -9130 -9110 -9090-9170 -9150 -9130 -9110 -9090

Depth

05*10^3

-9170 -9150 -9130 -9110 -9090-9170 -9150 -9130 -9110 -9090

Sep.2004

–p.51

Self-Organizing Maps (SOM)

A biologically motivated clustering method proposed byT. Kohonen ( 5 1981)

Viewed as a nonlinear projection of the probabilitydensity function % �rq �

of high dimensional input vectorq ] s t

onto the 2D plane

Neighboring points ins t

are mapped to neighboringpoints in

s @Also viewed as a vector quantization: approximate anyinput vector by the closest vector in a set of referencevectors

The Euclidean distance is often used. Sep. 2004 – p.52

Self-Organizing Maps (SOM) . . .

Consider a 2D array of nodes organized as ahexagonal lattice.

At a node located at u J , an initial reference (random)vector v J � Y � ] s t

is associated,

N & O� , , ,� w

.

A sequence of input vectors q � ' � ] s t

,

' & Y� O� , , , arecompared with the reference vectors

x v J � ' �y

.

For each

'

, the “best-matching” node

� u{z� vz � ' � �

withq � ' �

is found:m &= | } ~ � ?W� J� � � q � ' � ) v J � ' � ��

� q � ' � ) vz � & ~ � ?W� J� � � q � ' � ) v J � ' � � ,

Sep. 2004 – p.53


Then, the neighboring nodes are updated by thefollowing rule (learning process):

v J � ' � O � & v J � ' � � �z J � ' �� q � ' � ) v J � ' ��

where

�z J � ' �

is called the neighborhood function, anon-negative smoothing kernel over the lattice points.

Normally,

�z J � ' � & � � � uz ) u J �� ' �

with

� � `� ' �� Y

as' � �, � �� ` �� Yas � � �.

Sep. 2004 – p.54


Typical examples of

�z J � ' �

:

�z J � ' � & � � ' �� ) � uz ) u J � @b P�� @ � ' � ��

or �z J � ' � & � � ' �� $� �� N ��

where� � ' �

is a learning-rate factor

Y c � � ' � c O

, � � ' �

isan effective width of the kernel, and

wz � ' �

is aneighborhood set of the best node m, all of which arenon-increasing functions of

'

.

Plasticity . . .

Sep. 2004 – p.55


Sep. 2004 – p.56

Self-Organizing Maps (SOM): AnExample

Sep. 2004 – p.57

Facies Classification from WaveletFeatures

Feature vectors where the ground truth of facies wereobtained from thin sections were fed to the trainedSOM, which resulted in the labels (G, P, M, W).

Clusters corresponding Grainstone-Packstone (grainsupported) and Mudstone-Wackstone (mud supported)were identified.

G G G

P G P M W

P G G W W M

P W M Sep. 2004 – p.58

Facies Class

courtesy: Bureau of Economic Geology: UT AustinSep. 2004 – p.59

Remarks on SOM

- Many parameters to be specified, and theperformance critically depends on some of them.

- Precise mathematical analysis of convergence istough (proof done only for 1D array of nodes).

+ Each iteration is computationally fast.

Sep. 2004 – p.60

Challenges

Apply “Diffusion Maps” to the data!

Issue of normalization of different measurements

Separation of environmental effects and truegeophysical properties of rocks and formations frommeasurements (the uncertainty principle!)

Sep. 2004 – p.61

Summary

Reviewed three geophysical problems

Detection and description of geometry followed bycharacterization of texture

Scale of Measurements: vast range

Dimensions of Measurements: high

Classical techniques have been used

Can significantly improve via new techniquesdiscussed in this IPAM MGA program

May be able to provide data if interested

Sep. 2004 – p.62

References

W. S. Harlan, J. F. Claerbout, and F. Rocca: “Signal/noiseseparation and velocity estimation,” Geophysics, vol.49,pp.1869–1880, 1984.

N. N. Bennett, R. Burridge, and N. Saito, “A method to detect andcharacterize ellipses using the Hough transform,” IEEE Trans.Pattern Anal. Machine Intell., vol. 21, no. 7, pp.652–657, 1999.

N. Saito, N. N. Bennett, and R. Burridge, “Methods of DeterminingDips and Azimuths of Fractures from Borehole Images,” USPatent Number 5,960,371, Grant Date: 9/28/99.

N. Saito, A. Rabaute, and T. S. Ramakrishnan, “Method forInterpreting Petrophysical Data,” UK Patent Number GB2345776,Grant Date: 1/16/01.

Sep. 2004 – p.63

References on Radon Transforms

G.Beylkin: “Discrete Radon Transform,” IEEE Trans. Acoust.,Speech, Signal Processing, vol.ASSP-35, no.2, pp.162–172,1987.

S.R.Deans: The Radon Transform and Some of Its Applications,Wiley Interscience, 1983.

I.M.Gel’fand, M.I.Graev, and N.Ya.Vilenkin: GeneralizedFunctions, vol.5, Integral Geometry and Representation Theory,Academic Press, 1966.

L.A.Santaló: Integral Geometry and Geometric Probability,Addison-Wesley, 1976.

L.A.Shepp and J.B.Kruskal: “Computerized tomography: The newmedical x-ray technology,” Amer. Math. Monthly, vol.85,pp.402–439, 1978.

Sep. 2004 – p.64

References on Hough Transform

J.Illingworth and J.Kittler: “A survey of the Hough transform,”Comput. Vision, Graphics, Image Processing, vol.44, pp.87–116,1988,

V.F.Leavers: “Which Hough transform?” CVGIP: ImageUnderstanding, vol.58, pp.250–264, 1993, This article cites 173articles!

J.Princen, J.Illingworth, J.Kittler: “A formal definition of the Houghtransform: Properties and relationships,” J. Math. Imaging andVision, vol.1, pp.153–168, 1992.

Sep. 2004 – p.65

References on SOM

T. Kohonen: Self-Organizing Maps, Springer-Verlag, 1995.

Sep. 2004 – p.66

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