Jean-Laurent Mallet. Geomodeling

238

CHAPTER

5.

ELEMENTS

OF

DIFFERENTIAL

GEOMETRY

Figur e

5.19

Wave

front

W(t)

and

associated

ray

path

r(t) crossing

an

interface

Ji

between

two

homogeneous

regions

71

and

'R,.

Considering

a

Dirac

impulse

at

source

point

r(0)

at

time

t

=

0,

then:

• the

surface

W(t)

corresponding

to the set of

points

of the 3D

space

hit by

the

signal

at

time

t > 0 is

called

a

"wave

front"

at

time

t > 0. As

suggested

in

figure

(5.19),

in the

following

discussion

it

will

be

assumed

that

W(t)

is

represented

by a

parametric regular equation

w(u,

v\t).

• the

parametric curve between

a

point

r(t)

on

W(t)

and a

point

r(t + At) on

W(t

+

At), where

r(t + At) is the

closest point

of

W(t

+ At) to

r(t),

is

called

a

"ray path,"

or

more simply

a

"ray."

This

implies

that

the

unit

tangent

t(t)

to the ray

path

at

point

is

identical

to the

unit

normal

vector

N(t)

=

N(w,

v t) to

W(i)

at

that

point.

This

suggests

choosing

the

parameterization

*w(u,v\i)

as

follows:

•

choose

a

unit vector U(t) orthogonal

to

t(t)

and

build V(t)

in

such

a way

that

(U(t),

V(t),

t(t)}

becomes

a

direct orthogonal

frame

which

implies

that

•

(re)parameterize

the

surface

W(t)

in the

neighborhood

of

r(t)

in

such

a way

that

Ray

crossing

an

interface

Let

us

consider

a

surface

H

denning

the

interface

between

two

homogeneous

regions

71

and

K

of the 3D

space.

Let us

assume

that

the

seismic

velocity

5.6.

EXAMPLES

OF

APPLICATIONS

TO

GEOLOGY

239

Figure

5.20

An

example

of

seismic rays launched

from

a

shot

point

S

then

traveling across

the 3D

regions

(layers)

of

the

geological model represented

in figure

(2.1):

the

rays

are

reflected

and

refracted

by the

horizons according

to

Snell's law.

V(x,

y,

z]

is

constant,

but

different,

in

these

two

regions

71,

and

71

and let a

and a be the

associated "slownesses"

defined

by

As

suggested

in

figure

(5.19),

let us

consider

a

particular

ray

path

crossing

H

at

point

p. In

this particular case,

it is

well

known

that

the ray

path

is

linear

in

7£,

and

7£

and the

following

notations

will

be

adopted:

r(i)

=

equation

of the ray in

71

f(t)

=

equation

of the ray in

72.

The

linearity

of the ray

path

in

71

and

7£

guarantees

that

the

unit tangent

to

this

path

is

constant

in

both regions:

t(i)

= t

=

unit tangent

in

'R,

t(i)

= t =

unit tangent

in

71

Let

N*

be the

unit normal vector

to the

interface

7i

at the

impact point

p

with

the ray

path. Snell's

law

states

that

the

following

relationship between

t and t is

always true:

Figure

(5.20)

is an

example

of

seismic rays launched

from

a

source point

S in

the

geological model represented

in

figure

(2.1).

Wave

front

crossing

an

interface

As

shown

in figure

(5.21),

the

curvature

of an

interface

7i

between

two

regions

71

and

71

having

different

constant

slownesses

a and a

modifies

the

shape

of

240

CHAPTER

5.

ELEMENTS

OF

DIFFERENTIAL GEOMETRY

Figur e

5.21

Modification

of the

geometry

of a

wave

front

W(t)

crossing

an

interface

Ti..

the

wave

front

on

both

sides

of the

interface. Depending

on the

curvature

of

the

wave

front,

the

density

of

energy carried

by

this

wavefront

will

tend

to

increase

or

decrease.

For

this

reason,

it is

important

to

determine

how the

curvature

of the

interface

modifies

the

curvature

of the

wave

front.

Let

p be the

impact point

of a ray

with

the

interface

H,

and let

(U*,

V*)

be the

pair

of

unit vectors located

in Tp

(H)

and

denned

by

Let

h(w*,

v*)

be the

parametric representation

of

Ti,

in the

neighborhood

of p

such

that

Let

L

w

,

L

w

,

and

L

h

be the

2x2

matrices corresponding

to the

second

funda-

mental tensors

of

W(t),

VV(t),

and

Ti.

at the

impact point

p. It can be

shown

(see

[110,

42])

that

these three matrices

denning

the

curvature

of

W(i),

VV(i),

and

Ti.

at the

impact point

are

linked

by the

following

matrix equation:

where

A,

F,

and

F

are

defined

by

Note

that

equation (5.78) plays

a

similar role

for the

curvature

of the

wave

front

W

as

Snell's

law

(see equation

(5.77))

for its

associated unit normal

vector

t(t).

5.6.

EXAMPLES

OF

APPLICATIONS

TO

GEOLOGY

241

Figure

5.22

Evolution

through

time

of

a

surface

element

on

W(t).

Expansion

of the

wave

front

Assuming

(w

u

,w^,N)

to be an

orthonormal

frame

and for

simplicity's sake,

the

following

notations

will

be

used

in

this section:

As

suggested

in

figure

(5.22),

let

p

0

(t),

Pi(t),

and

p

2

(t)

be

three points

on

the

tangent plane

to

W(t)

such that

These three points

are

assumed

to

move through time

on ray

paths, and,

at

time

t +

At,

their locations

are

given

by

where

As = V •

At

is the

length

of

displacement along

the ray

path

at

velocity

V

during

the

increment

of

time

At.

Let

a(t)

and a(t + At) be the

vectors

defined

at

times

t and t + At by

It is

clear

that

the

modules

||a(t)|

and

\\a(t

+ At) of

these

vectors measure

twice

the

areas

of the

triangles

denned

by the

points

p

0

,

p

1

and

p

2

at

times

t and t + At,

respectively.

We

propose calling "expansion factor"

of

W(t)

at

location -w(u,v\t) the function (f)(u,v\t,At,V) which depends on the seismic

242

CHAPTER

5.

ELEMENTS

OF

DIFFERENTIAL GEOMETRY

velocity

V and is

defined

by

14

We

will

now

show

that

this expansion factor depends solely

on the

curvature

of

the

wave

front

W(t)

at

location

\v(u,v

t}.

By

definition

of a(t + At)

According

to

equation

(5.55),

where

the

coefficients

{Ifj

(t)}

represent

the

components

of the

second

funda-

mental tensor

L

w

(t)

of

VV(i)

at

location

vf(u,v\t).

The

following

implication

is

easily

verifiable:

Equations (5.81),

and

(5.53) lead

us to

conclude

that

where

H

w

(u,v\t]

and

K

w

(u,

v\i)

represent

the

mean

and

total

curvatures (see

equation

(5.52))

of the

wave

front

W(t)

at

location

w(u,

v\t),

respectively.

This implies

that

the

expansion factor

0(it,

v\t,

At,

V"),

as

defined

by

equation

(5.80),

is

such

that

15

It is

interesting

to see

that, similar

to

H

w

(u,

v\t)

and

K

w

(u,

v\i),

the

expansion

factor

0(w,

v\t,

At, V) is

independent

of the

parameterization

of the

wave

front

W(t).

In

practice, this expansion factor

can be

used

for

computing

the

variation

of

the

density

of

energy

per

unit

of

surface carried

by the

wave

front

along

a ray

from

point

w(w,v|t)

to

point

w(u,v\t

+ At) at

velocity

V.

If we

note

14

This

definition only makes sense

if

a.(t)

and

a(t

+ At) are

approximately colinear.

15

It

can be

noted

that

if set x =

I/As

then

equation (5.54) multiplied

by

l/x

2

becomes

identical

to the

right-hand

side

of

equation

(5.83).

5.6.

EXAMPLES

OF

APPLICATIONS

TO

GEOLOGY

243

E(u,v\t)

this density

of

energy

at

point

w(u,v\t)

on the

wave

front,

then

E(u,

v\t

-f

At)

at

point

w(u,v\t

+ At) is

such

that

For

this

reason,

the

function

l/(f)(u,v\t,

At,

V) is

called

the

"spreading

/actor"

of

the

energy along

the ray

path.

Comment

From equation

(5.52),

it can

also

be

observed

that

equation

(5.82)

can be

fac-

torized into

the

following

form

where

k™(u,

v\t)

and

k%(u,

v\t)

are the

principal

curvatures

of

W(t)

at

location

w(u,

v

t):

5.6.4

Fluid

flow

through

porous

media

The

velocity v(x)

of a fluid

particle

at a

point

x in a

porous media

is

governed

by

Darcy's

law

which,

in the

case

of

steady

flow, can be

written

as

In

this equation,

(p

is the fluid

pressure, while

IK is a

(twice contravariant)

tensor called

"permeability

tensor""

1

and is

represented

by a 3 x 3

symmetric

matrix:

In the

isotropic

case,

IK

reduces

to the

following

form

In

this case, equation

(5.84)

clearly shows

that

v is

colinear

to

grad</?.

In

other words,

in the

isotropic case,

the flow

velocity

v is

orthogonal

to the

isopressure

surfaces,

and

there

is a

perfect

analogy with

ray

tracing:

•

curves

tangent

to v are

called

"stream

lines"

and are the

analog

of ray

paths,

and

•

isopressure

surfaces, whose implicit

equation

is

<^(x)

=

C

te

,

are the

analog

of

the

wave fronts.

As

a

consequence,

stream

tubes

corresponding

to

bunches

of

adjacent

stream

lines have

a

section

that

is

ruled

by an

equation similar

to

equation

(5.82).

Let

us

parameterize

the

stream

lines

by

their

arc

length

s.

From equation

(5.82),

we

deduce

that

section a(s)

of

stream tubes evolves according

to the

following

equation

244

CHAPTER

5.

ELEMENTS

OF

DIFFERENTIAL GEOMETRY

where

H(s)

and

K(s)

represent

the

mean

and

total

curvatures

of the

isopres-

sure surfaces

at

point

x(s)

on the

stream line corresponding

to the

axis

of the

stream tube, respectively.

5.7

Conclusions

The

basic principles

of

differential

geometry

of

curves

and

surfaces have been

introduced

in

this chapter

to

allow

a

better understanding

of

chapters

6 and

7

dedicated

to the

modeling

of

surfaces.

In

particular,

it

should

be

noted

that

•

chapter

6

makes

intensive

use of the

notions

metric

tensor

and

curvatures

of

parametric

surfaces,

and

•

chapter

7

relies

on the

notion

of

G

n

continuity

of

parametric

surfaces

to

build

composite

smooth

surfaces

consisting

of a

patchwork

of

curvilinear

triangular

patches.

Moreover,

a few

direct applications

of

differential

geometry

to

geological prob-

lems

have been presented

in

section 5.6; these

specific

applications cover

a

wide

diversity

of fields

such

as the

modeling

of

channels,

the

detection

of

faults

or

the

propagation

of

seismic wave

fronts

and

stream tubes

in the

subsurface.

Chapter

6

Piecewise Linear

Triangulated Surfaces

From

a

topological

and a

geometrical

point

of

view,

an

earth model consists

of

a

partition

of the 3D

geological space

into

homogeneous regions bounded

by two

types

of

surfaces:

•

horizons, which correspond

to

interfaces

between layers and/or fades

generated,

for

example,

by

sedimentological events,

and

•

faults, which correspond

to

discontinuities generated

by

tectonic events.

Therefore,

surfaces

play

a

central role

in the

modeling

of

the

subsurface.

This

chapter

shows

how

DSI

can be

used

for

modeling these

surfaces,

while taking

into account

a

wide variety

of

constraints.

In

practice,

for

simplicity's sake

and

as

suggested

in figure

(6.1),

it

will

be

assumed that

surfaces

are

decomposed

into

triangular

fiat

facets.

6.1

Introduction

Building complex geological models with heavy tectonic deformations requires

tools able

to

handle

not

only

the

geometry

of

faults

and

geological horizons

correctly,

but

also

the

relationships between these objects.

As

shown

in

this

chapter,

it is

possible

to

design

DSI

constraints

that

take into account

the

intrinsic nature

of

these surfaces, while preserving their relationships.

To

simplify

the

construction

of

such complex constraints,

this

section

presents

a set of

consistent definitions

and

notations

that

will

be

used through-

out

this chapter.

245

246

CHAPTER

6.

PIECE

WISE LINEAR TRIANGULATED SURFACES

Figure

6.1

An

example

of

a

piecewise

linear

triangulated

surface

(salt

dome).

6.1.1

Notion

of a

triangulated

surface

Definitions

Let

t$

be a

surface.

By

definition,

a

triangulated surface associated with

<5

is

any

cellular partition

P(S]

of S

where each

2-cell

(see section

(2.3.1))

is a

2-simplex,

that

is to

say,

a

triangle.

For the

sake

of

clarity, such

a

partition

will

be

noted

as

T(«S)

rather than

7^(5)

throughout this chapter.

For any

triangulated surface

T(<S)

considered within

the

framework

of

this

chapter, G(fi>T(S)i A^s)) and •Mn(^T(S)^T(S)^ f-, £</>) are defined as follows:

•

G(£IT(S),

NT(S))

is the

2-dimensional

graph (see section

(1.2))

associated with

T(5)

and

—

QT(S)

—

se

t

of all the

vertices

of

T(«S),

—

Af<7-(s)(oO

— set of all the

vertices

(3

G

QT(S)

such

that

(ct,/3)

is an

edge

of

T(S)

•

-M

n

(r27-(S),

-^Vr(5)i

V

3

)^)

is a

discrete model (see section

(1.2))

associated with

T(<S)

and

corresponding

to the

modeling

of a

vectorial property

denned

for any

node

a of

QT(S)

•

In

practice,

for

simplicity's sake,

in the

rest

of

this chapter,

the

subscript

T(S]

will

be

removed

from

fi^s)

an

d

NT(S)

an

d

the

following

notations

will

be

used:

6.1.

INTRODUCTION

Geometr y

and

global

parameterization

of a

surface

As

discussed

in

section

(5.2),

any

mapping

1

x

from

a 2D

(parametric) domain

D

onto

a

surface

S

is

called

a

"global parametric

representation"

of S or,

more

simply,

a

parametric surface:

In the

rest

of

this chapter,

it

will

be

assumed

that

<S

belongs

to 3D

space

7R

3

,

and

x(u)

has

three components noted

as

follows:

For

simplicity's sake,

the

following

notations

will

be

used:

In

association with these notations,

the two

following

discrete models describ-

ing

the

sampling

on

O

of the

global parameterization

u and the

geometry

x

of

the

surface

<S

will

also

be

considered:

•

Ai

3

(0,TV,

x,

Cx)

models

the

sampling

on

O

of the

geometry

of S in the

embedding space

JR

3

,

and

•

A^

2

(fi,

TV,

u,

Cu)

models

the

sampling

on £1 of the

global parameteriza-

tion

u of

£

in the

parametric space

IR

2

.

Property

attached

to a

surface

Let

v?(u)

=

[(/?

1

(u),..

.,9?

n

(u)]*

be a

vectorial

function

defined

on the

same

parametric domain

D as a

parameterization x(u)

of a

surface

S:

As

with

x and u,

this property

will

be

assumed

to be

sampled

on

Q

and

associated with

a

discrete model

A4

n

(Q,

TV,

</?,

C

v

}.

In

geology,

(p

represents (physical) properties

of a

geological surface

S

such

as, for

example,

a

seismic

reflectivity

or the

thickness

of the

layer located

on

a

given side

of S.

Triangular

facet

In the

rest

of

this book,

T =

T(X(Q!O),

x(a;i),

x(oi2))

will

represent

a

triangular

facet

belonging

to a

triangulated

surface

T(S}:

(ao)

«i)

is an

edge

of

T(S]

(QI,

0:2)

is an

edge

of

T(<5)

(0:2,

cto)

is an

edge

of

T(S]

1

Note

that

such

a

global parameterization

is not

unique.

247

Jean-Laurent Mallet. Geomodeling

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