Russell - Introduction to Seismic Inversion Methods

Introduction

to

Seismic Inversion Methods

Brian H. Russell Hampson-Russell SoftwareServices,Ltd. Calgary,Alberta

Course Notes Series, No. 2 S. N. Domenico, Series Editor

Societyof Exploration Geophysicists

Thesecoursenotesare publishedwithoutthe normalSEGpeerreviews. They havenot beenexaminedfor accuracyand clarity.Questionsor commentsby the readershouldbe referreddirectlyto the author.

ISBN 978-0-931830-48-8 ISBN 978-0-931830-65-5

(Series) (Volume)

Libraryof Congress CatalogCardNumber88-62743 Societyof Exploration Geophysicists P.O.

Box 702740

Tulsa, Oklahoma 74170-2740

¸ 1988 by the Societyof Exploration Geophysicists All rightsreserved. Thisbookor portionshereofmaynotbe reproduced in anyformwithoutpermission in writingfromthe publisher. Reprinted1990, 1992, 1999, 2000, 2004, 2006, 2008, 2009 Printed in the United States of America

]:nl;roduct1 on •o Selsmic I nversion •thods

Table

Bri an Russell

of Contents PAGE

Part

I

Introduction

1-2

Part

Z

The Convolution

Model

2-1

2.4 The Noise Component

2-2 2-6 2-12 2-18

Recursive

3-1

2.1 Tr•e Sei smic Model 2.2 The Reflection Coefficient 2.3

Part

Part

3

4

The Seismic

Wavelet

Inversion

- Theory

3.1

Discrete

3.2

Problems encountered

Inversion

3.3

Continuous

with

data

4-1

4. !

4-2 4-4 4-6 4-12 4-14

4.4 4.5

I ntroduc ti on resolution

Lateral resolution Noise attenuation

Recursive

Inversion

- Practice

5.3 Seismically derived porosity Sparse-spike Inversi on

6-1

6.1

6-2 6-4 6-22 6-30

The recursive

inversion

method

5.2 Information in the low frequency component 6

I ntroduc ti on

6.2 Maximum-likelihood 6.3 6.4 P art

Part

7

8

5-1 5-2 5-10 5-16

5.1

P art

3-2 3-4 3-8

Seismic Processing Considerati ons 4.2 Ampli rude recovery

5

real

Inversion

4.3 Improvementof vertical

Part

Series

aleconvolution and inversion

The L I norm method Reef Problem

I nversion applied to Thi n-beds

7-1

7.1 Thin bed analysis

7.Z Inversion comparison of thin beds

7-2 7-4

Model-based

8-1

Inversion

B. 1 I ntroducti

8.2

Generalized

on .

linear

inversion

8.3 Seismic1ithologic roodelling (SLIM) Appendix8-1 Matrix applications in geophysics

8-2 8-4 8-10 8-14

Introduction

Part

9

to Seismic

Travel-time

Inversion

Methods

Brian

9-1

Inversion

g. 1. I ntroducti

on

9.2 Numerical examplesof traveltime 9.3 Seismic Tomography

inversion

Part 10 Amplitude versus offset (AVO) Inversion 10.1 AVO theory 10.2 AVO inversion by GLI

9-2 9-4 9-10

10-1 10-2 10-8

Inversion

11-1

I ntroduc ti on

11-2 11-4

Part 11 Velocity

Theory and Examples Part 12 Summary

12-1

Russell

Introduction

to Seismic •nversion

Methods

Brian Russell

PART I - INTRODUCTION

Part

1 - Introduction

Page 1 -

1

Introduction

to Seismic

Inversion

Methods

Brian

Russell

I NTRODUCT ION TO SEI SMIC INVERSION METHODS ,

__

_•

i

Part

This

i

_

i

,

.

,

-

,

!

•

_,

l_

,

Introduction _

,

.

i.,.

_

.

course is intended as an overview of the current techniques used in

the inversion of seismic data. It would therefore seemappropriate to begin by defining what is meant by seismic inversion. The most general definition is

as fol 1 ows'

Geophysical inversion

involves

mapping the physical structure and

properties of the subsurface of the earth using measurementsmadeon the surface of the earth.

The above definition

work that is done in

is so broad that it encompassesvirtually

seismic analysis

course we shall primarily 'restrict

and interpretation.

all

the

Thus, in this

our discussion to those inversion

methods

which attempt to recover a broadband pseudo-acoustic impedance log from a band-1 imi ted

sei smic trace.

Another way to look at inversion is to consider it as the technique for creating a model of the earth using the seismic data as input. As such, it

can be considered as the opposite of the forwar• modelling technique, which involves creating

a synthetic seismic

section

based on a model of the earth

(or, in the simplest case, using a sonic log as a one-dimensional model). The relationship between forward and inverse modelling is shownin Figure 1.1.

To understandseismic inversion, we must first processes involved therefore

in

the

look at the basic

creation of seismic data. convolutional

model

understandthe physical Initially,

we will

of the seismic trace

in the

time and frequencydomains,consideringthe three components of this model: reflectivity,

seismic wavelet, and noise.

_

Part

I

- Introduction

m

i

Page 1 -

--.

2

Introduction to Seismic InverSion Methods

FORWARDMODELL I NG i

m

Brian Russell

INVERSEMODELLING(INVERSION) _

,

ß

ß

_

ß

EARTH MODEL

Input'

,

Process:

Output'

MODELLING

INVERSION

ALGORITHM

ALGORITHM

SEISMIC RESPONSE i

m

mlm

ii

EARTH MODEL i

ii

Figure1.1 Fo.•ard ' andsInverse Model,ling

Part

I - Introduction

Page I -

3

Introduction.

to Seismic Inversion Methods

Brian l•ussel 1

Once we have an understanding of these concepts and the problems which

can occur, we are in a position to

look at

the methodswhich are currently

ß

used to invert seismic data.

These methods are summarizedin Figure 1.2.

be on poststack seismic inversion where

primary emphasis of the course will the

ultimate

resul.t,

The

o

as was previously

Oiscussed, is a

pseudo-impeaance

section.

We will

start by looking at

inversion,

the

most contanonmethods of

which are based on single trace recursion.

these recurslye relationship

inversion

procedures,

it

the

unUerstand

important to look at the

between aleconvolution anU inversion, and how Uependent each

method is on the deconvolution scheme Chosen.

classical

is

To better

poststack

Specifically,

we will

consider

"whitening" aleconvolutionmethods, wavelet extraction methods, and

newer sparse-spike deconvolution methods such as Maximum-likelihood

deconvolution

and the

L-1

norm metboa.

Another important type of inversion methodwhich will be aiscussed is model-based inversion, where a geological moael is iteratively upUatedto finU the

best

fit

with the seismic data.

After this,

traveltime

tomography,will be discussedalong with several illustrative

inversion,

or

examples.

After the discussion on poststack inversion, we shall move into the realm of pretstack. These methoUs,still fairly new, allow us to extract parameters other than impedance, such as density and shear-wave velocity. Finally,

we will

aiscuss the geological aUvantages anU limitations

each seismic inversion roethoU,looking at examples of each.

Part

1 -

Introduction

Page i -

of

Introduction to SelsmicInversion Methods

Brian Russell

SEI SMI C I NV ERSI ON

.MET•OS,,,

POSTSTACK

PRESTACK

INVERSION

INVERSION

i

EF IEL D MODEL-BASED I RECURSIVEWAV TRAVELTIME

INVERSION • ,INVESION

INVERSION

,,

I METHODS ]

!TOMOGRAPHY) - "NARROWSPARSEBAND

Figure 1.2

Part

1 - Introuuction

--

LINEAR

NVERSIOUMETHODS i

i

SPIKE

A summaryof current inversion techniques.

Page 1 -

Introduction

to Seismic Inversion Methods

Brtan Russell

PART 2 - THECONVOLUTIONAL MODEL

Part

2 - The Convolutional

Model

Page 2 -

Introduction

to Seismic Inversion Methods

Part 2 -

Brian Russell

The Convolutional

Mooel

2.1 Th'e Sei smic Model

The mostbasic and commonly used one-Oimensionalmoael for the seismic trace is referreU

to as the convolutional

moOel, which states that the seismic

trace is simplythe convolutionof the earth's reflectivity with a seismic source function

with the adUltion of a noise component. In

equation

form,

where * implies convolution,

s(t) : w(t) * r(t) + n(t)s where

and

s (t)

= the sei smic trace,

w(t)

: a seismic wavelet,

r (t)

: earth refl ecti vi ty,

n(t)

: additive

noise.

An even simpler assumptionis to consiUerthe noise component to be zero, in which case the seismic tr•½e is simply the convolution of a seismic wavelet with t•e earth ' s refl ecti vi ty, s(t)

In

= w{t) * r(t).

seismic processingwe deal exclusively with digital data, that

data sampled at a constanttime interval.

is,

If weconsiUerthe relectivity to

consist of a reflection coefficient at each time sample(som• of which can be

zero), and the wavelet to be a smooth function in time, convolutioncan be thoughtof as "replacing"eachreflection. coefficient with a scaledversion of the waveletandsumming the result. The result of this processis illustrated

in Figures 2.1 and2.Z for botha "sparse"anda "dense"set of reflection coefficients.

Notice that convolution

with

the wavelet tends to "smear" the

reflection coefficients. That is, there is a total loss of resolution,which is the ability

to resolve closely spacedreflectors.

Part 2 - The Convolutional

Model

Page

Introduction

to Seismic Inversion

Nethods

Brian Russell

WAVELET:

(a) '*• •

:

-' ':'

REFLECTIVITY

TRACE:

Figure 2.1

(a)

Convolutionof a wavelet with a sparse"reflectivity. (a) •avelet. (b) Reflectivit.y. (c) Resu1ting Seismic Trace.

!

.

i

:

!

!

:

i

,

:

i

!

i

'?t

*

(b')

:

i

ß

i

c

o

Fi õure 2.2

o

Convolution of a wavelet with a sonic-derived

reflectivity. ,

Par•

i

o

, ß ....

!

,

m

2 - The Convolutional

(a) Wavelet. (b) Reflectivity. i

i

L_

Model

-

o

o

"dense"

(c) SeismicTrace

'

Page 2 -

3

Introduction

to Seismic

An alternate,

Inver'sion

Methods

Brian

but equivalent, way of

the frequency domain.

Russell

looking at the seismic trace is in

If we take the Fourier transform of

the previous

ß

equati on, we may write

S(f) where

= W(f) x R(f),

S(f) = Fouriertransform of s(t), W(f) = Fourier transform of w(t), R(f) = Fourier transform of r(t),

In the above equation we see that

ana f = frequency.

convolution becomesmultiplication

in

the frequency domain. However, the Fourier transform is a complex function, and it

is normal to consiUer the amplitude and phase spectra of the individual

components. The spectra of S(f) may then be simply expressed

esCf)= ew(f) + er(f), where

I •ndicates amplitude spectrum, and 0

In the

Figure 2.3

illustrates

the convolutional model

frequency domain. Notice that the time Oomainproblem of

resolution becomesone of loss of reOuceo by the effects

2 - The Convolutional

loss

of

frequency content in the frequency domain.

Both the high and low frequencies of the reflectivity

Part

.

other words, convolution involves multiplying the amplitude spectra

and adding the phase spectra.

in

indicates phase spectrum.

have been severely

of the seismic wavelet.

Mooel

Page ?. -

4

Introduction to Seismic Inversion Methods

AMPLITUDE

Brian Russell

SPECTRA

PHASE SPECTRA

w (f) I

I

-tR (f)

i i

, i.

I iit

!

loo

|11

s (f) i

I

i

i!

I

Figure 2.3

Part

2 - The Convolutional

Convolution in the frequency domain for the time series shown in Figure 2.1.

Model

Page 2 -

Introduction

2.g

to Seismic

The Reflection l_

_

,m

i

_

_

,

Inversion

Coefficient _

_

m_

_,•

,

Methods

Brian

Russell

Series _ _

ß

_

el

'The reflection coefficient series (or reflectivity,

as it is also called)

describedin theprevious sectionis oneof thefundamental physical concepts in the seismic method. Basically, each reflection coefficient maybe thought of as the res ponse of the seismic wavelet to an acoustic impeUance change within the ear th, where acoustic impedance is defined as the proUuct of compressi onal velocity and Uensity. Mathematically, converting from acoustic involves dividing the difference in the acoustic i ropedanceto re flectivity impedances by the sum of the acoustic impeaances. This gives t•e coefficient

at

the boundary between the two layers.

reflection

The equation is as

fo11 aws:

•i+lVi+l- iVi i where

•

Zi+l- Zi i+1

r = reflection

coefficient,

/o__density, V -- compressional velocity, Z -- acoustic impeUance, and

Layer i overlies Layer i+1.

Wemust also convert from depth to time by integrating the sonic log transit times. Figure •.4 showsa schematicsonic log, density log, anU resulting acoustic impedancefor a simplifieU

earth moael. Figure 2.$ shows

the resultof converting to thereflection coefficient seriesandintegrating to

time.

It should be pointed out that this formula is true only for the normal incidence case, that is, for a seismic wave striking the reflecting interface at right angles to the beds. Later in this course, we shall consider the case of

Part

nonnormal

inciaence.

2 - The Convolutional

Model

Page 2 -

6

Introduction to Seismic Inversion Methods

STRATIGRAPHIC

Brian Russell OENSITY

SONICLOG

SECTION

30O

4OO

loo

200

2.0

3.0

,

ß

I

SHALE

LOG.

•T (•usec./mette)

.....

•

3600 m/s

DEPTH

_

SANOSTONE

ß

ß ß

ß

. .

'I

!

- .. ,

!_1

!

ß ß

!

v--I

!

UMESTONEI I I ! I ! I 1

V--3600 J

V= 6QO0

LIMESTONE

2000111

I

Fig. 2.4. BoreholeLogMeasurements. REFLECTWrrY

ACOUSTIC

VS TWO.WAY TIME

IMPED,M•CE (2•

(Y•ocrrv mm

,mm

mm

mm

rome

m

m

-----

V$ OEPTH

x OEaSn• 20K -.25 I

.am

O

Q.2S I

-.25 v

O '

+ .2S I

mm

SHALE .....

OEPTH

•--------'-[

SANDSTONE . . ... !

I

!11

,

I1

UMESTONE I I 1 I I I II i ! I 1 i I i SHALE •.--._--.---- •

1000m

-

1000 m

--

NO

•.'•

,•

LIMESTONE

- 20o0 m

2000 m

Fig.

2.5.

, ..

Creation of Reflectivity

Part g - The Convolutional Model

I SECOND

Sequence.

Page 2 -

7

IntroductJ

on 1:o Sei stoic Inversion

Our best method of

derlye

Herhods

observing

them from well log curves.

Bri an Russell

seJsm•c impedance and reflectivity

is •o

Thus, we maycreate an impedancecurve

by

multiplying together •he sonic and density logs from a well. Wemay•hen computethe reflectivlty by using •he formula shownearlier. Often, we do not have the density log available• to us and must makedo with only the sonJc. The

approxJmatJonof velocJty to •mpedance 1s a reasonable approxjmation, and

seemsto holdwell for clas;cics and carbonates(not evaporltes, however). Figure 2.6 showsthe sonic and reflectJv•ty traces from a typJcal Alberta well after they have been Jntegrated to two-way tlme. As we shall see later,

the type of

aleconvolution and inversion used is

dependent on the statistical assumptionswhich are made about the seismic reflectivity and wavelet. Therefore, howcan we describe the reflectivity seen in

a

well?

reflectivity

The

traditional

to be a perfectly

answer has always been that

we consider

random sequence and, from Figure •.6,

the

this

appears to be a good assumption. A ranUomsequencehas the property that

autocorrelation is a spike at zero-lag. autocorrelation are zero except the

its

That is, all the componentsof the

zero-lag value, as shownin the following

equati on-

t(Drt = ( 1 , 0 , 0 , .........

)

t zero-lag.

Let

2.7.

us test this idea on a theoretical

Notice that the

autocorrelation

of

random sequence, shownin

Figure

this sequence has a large spike at

ß

the zeroth lag, but that there is a significant noise component at nonzero lags. To have a truly random sequence, it must be infinite in extent. Also on this figure is shown the autocorrelation of a well log •erived

reflectivity. Wesee that it is even less "random"than the randomspike sequence. Wewill discuss this in more detail on the next page.

Part

2 - The Convolutional

Model

Page 2 -

8

IntroductJon

to Se•.s=•c Inversion

Methods

Br•an

Russell

RFC

F•g. 2.6. Reflectivitysequence derivedfromsonJc .log.

RANDOM

SPIKE SEQUENCE

AUTOCORRE•JATION OF RANDOMSEQUENCE

Fig.

2.7.

WELL LOG DERIVED REFLECT1vrrY

AUTOCORRELATION

OF REFLECTIVITY

Autocorrelat4ons of random and well log

der4vedspike sequences.

Part

2 - The Convolutional

Model

Page 2-

Introductlon

to Sei smic Inversion

Methods

Therefore, the true earth reflectivity

truly

Brian Russel 1

cannot be consideredas being

random. For a typical Alberta well we see a numberof large spikes

(co•responding to majorlithol ogic change)sticking up abovethe crowd.A good way to describethis statistically is as a Bernoulli-Gaussian sequence. The Bernoulli part of this term implies a sparsenessin the positions of the spikes and the Gaussianimplies a randomness in their amplitudes. Whenwe generatesuch a sequence,there is a term, lambda, which controls the sparsenessof the spikes. For a lambdaof 0 there are no spikes, and for a lambdaof 1, the sequence is perfectly Gaussian in distribution. Figure 2.8 shows a number of such series for different

typical Alberta well log reflectivity

values of lambda.

Notice that

a

wouldhavea lambdavalue in the 0.1 to

0.5 range.

Part

2 - The Convolutional

Model

Page 2 -

10

I ntroducti on to Sei smic I nversi on Methods

Brian Russell

It

tl

I

I

I

i

I

I

•11 I

LAMBD^•0.01

511 t

•tl

I

(VERY SPARSE)

11

311

I

4#

I

511 I

#1

TZIIE

LAMBDA--O.

I

(KS !

1

1,1

::.•"• •'•;'"' "";'•'l•' "••'r'•

-• "(11 I TX#E

(HS)

LAMBDAI0.5

LAMBDA--

1.0 (GAUSSIAN:]

EXAMPLESOF REFLECTIVITIES

Fig.

2.8.

Examplesof reflectivities factor

,

,

m

i

ß

to be discussed

using lambda

in Part

6.

i

Part 2 - The Convolutional Model

Page 2 -

11

Introduction

2.3

to Seismic Inversion ,Methods

The Seismic --

_

Brian Russell

Wavelet

ß

•

,

Zero Phase and Constant Phase Wavelets m _

m _

m

ß

m

u

,

L

m

_

J

The assumptiontha.t there is a single, well-defined wavelet which is convolved with the reflectivity to producethe seismic trace is overly simplistic. Morerealistically, the wavelet is both time-varying and complex in shape. However,the assumptionof a simple wavelet is reasonable, and in this

section

we shall

consider

several

types

of

wavelets

and

their

characteristics.

First,

let us consider the Ricker wavelet, which consists of a peak and

two troughs, or side lobes. The Ricker wavelet is dependentonly on its dominant frequency, that is, the peak frequencyof its a•litude spectrum or the inverse of the dominantperiod in the time domain(the dominantperiod is

found by measuringthe time from troughto trough). TwoRicker wave'lets are shownin Figures 2.9 and 2.10 of frequencies 20 and 40 Hz. Notice that as the anq•litude spectrumof a wavelet .is broadened,the wavelet gets narrower in the

timedomain, indicatingan increase of resolution.Ourultimatewaveletwould be a spike, with a flat amplitude spectrum. Sucha wavelet is an unrealistic goal in seismic processing, but one that is aimedfor. The Rtcker wavelets of

Figures 2.9

and 2.10

are also zero-phase, or

perfectly symmetrical. This is a desirable character.tstic of wavelets since the energy is then concentrated at a positive peak, and the convol'ution of the wavelet

with

a reflection

coefficient

will

better

resolve

that

reflection.

To

get an idea of non-zero-phase wavelets, consider Figure 2.11, where a Ricker wavelet

has been rotated by 90 degree increments, and Figure 2.12, where the

samewavelet has been shifted by 30 degree increments.

Notice that the 90

degree rotation

180 degree shift

displays perfect antis•nmnetry, whereas a

simply inverts the wavelet.

Part

2 - The Convolutional

The 30 degree rotations are asymetric.

Model

Page 2-

•2

Introduction to Seismic Inversion Methods

Brian Russell

Fig.

2.9.

20 Hz Ricker

Wavelet'.

Fig.

•.10.

40 Hz Ricker

wavelet.

Fig.

2.11.

Ricker

wavelet

rotated

by 90 degree increments

Fig.

2.12.

Ricker

wavelet

rotated

by 30 degree increments

Part

2 - The Convolutional

Model

Page 2 -

13

Introduction

to Seismic

Of course,

Inversion

a typical

frequencies than that

fil•er

Methods

seismic wavelet

shownin Figure 2.13,

would be noticeable

a larger

Consider the

range of banapass

where we have passed a banaof frequencies has also had cosine tapers applied between 5

and 15 Hz, and between60 and 80 Hz. box-car.

contains

shownon the Ricker wavelet.

between15 and 60 Hz. The filter

that

Brian Russell

The taper reduces the "ringing" effect

if the wavelet amplitude spectrum was a

The wavelet of Figure 2.13 is

simple

zero-phase, and would be excellent as

a stratigraphic wavelet. It is often referred to as an Ormsbywavelet. Minimum Phase Wavelets

The concept of minimum-phaseis one that

is

vital

to aleconvolution, but

is also a concept that is poorly understood.

The reason for

understanding is that most discussions of

concept stress the mathematics

at

the

expense of

the

physical

the

interpretation.

use of minimum-phaseis adapted from Treitel

The

this lack of

definition

we

and Robinson (1966):

For a given set of wavelets, all with the sameamplitude spectrum,

the minimum-phase waveletis the onewhichhasthe sharpest leading edge. That is, only wavelets which have positive

The reason that

time values.

minimum-phase concept is important to us is

that

a

typical wavelet in dynamite work is close to minimum-phase. Also, the wavelet from the

seismic instruments

equivalent of the 5/15-60/80

in the aefinition

is

also

zero-phase wavelet is shownin Figure 2.14.

Part

As

used, notice that the minimum-phasewavelet has no component

prior to time zero and has its energy

possible.

minimum-phase. The minimum-phase

concentrated

as close to the origin

as

The phase spectrum of the minimum-waveletis also shown.

2 - The Convolutional

Model

Pa.qe 2 -

14

I•troduct•on to Seistoic!nversionNethods. ql

Re• R f1.38

Zero Phase I•auel•t

5/15-68Y88

-

0.6

- e.3e

{•

Trace

1

iii

...... ,

• .....

'

1

2be Trace

Reg 1)

Br•anRussell

min,l•

wavelet

Fig.

2.13. Zero-phase bandpass

Fig.

2.14. Minim•-phase equivalent of zero-phase wavelet shownin Fig. 2.13.

wavelet.

I

•/15-68/88 hz

18.00 p

Trace I

RegE

wayel

Speetnm

'188.88•

Trace1

0.8

188

_

!

m,m,

i

m

Part 2 -Th 'e Convolutional Model

i

Page 2-

15

Introduction

to Seismic Inversion Methods

Brian Russell

Let us nowlook at the effect of different waveletson the reflectivity function itself.

Figure 2.15 a anU b shows a numberof different

conv6lved with the reflectivity in Figure Z.5.

(Trace 1) from the simple blocky model shown

The following wavelets have been used- high frequency

zero-phase (Trace •),

low frequencyzero-phase(Trace ½), high frequency

minimumphase (Trace 3), figure,

wavelets

low frequency minimum phase (Trace 5).

From the

we can make the fol 1owing observations:

(1) Low freq. zero-phase wavelet: - Resolution of reflections

- Identification

(Trace 4) is poor.

of onset of reflection

is good.

(Z) High freq. zero-phase wavelet: (Trace Z) - Resolution of reflections

- Identification

is good.

of onset of reflection

is good.

(3) Lowfreq. min. p•ase wavelet- (Trace 5) - Resolution

of reflections

i s poor.

- Identification of onset of reflection is poor. (4) High freq. min. phase wavelet: (Trace 3) - Resolution of refl ec tions is good.

- Identification

of onset of reflection

is poor.

Based on the aboveobservations, we wouldhave to consider the high

frequency,zero-phase waveletthe best, andthe low-frequency, minimum phase wavelet

Part

the

worst.

2 - The Convolutional

Model

Page 2 -

16

Introduction

!ql

RegR

to Seismic Inversion

Zer• PhaseUa•elet

•,'1G-•1•

Methods

14z

Russell

q2 RegC ZeroPhase 14aue16(' ' •'le-3•4B Hz e

F

- •.• ['

Brian

'

•,3 RecjB miniilium phue

'

q• Reg1) 'minimum phase "

'

•,leJ3e/4eh•

'

8

17 .•

e.e

(a)

•/••/'•-•"v--,._,, -r

e.•' ' "s•e'' ,m

,,

Tr'oce

[b) 700

Fig.

2.15.

Convolution of four different in (a) with trace I of (b). shown on traces

Part

2 - The Convolutional

Model

wavelets shown The results are

2 to 5 of (b).

Page 2 -

17

Introduction

to Seismic Inversion

Methods

Brian Russell

g.4 Th•N.oi se.Co.mp.o•ne ntThe situation

that has been discussed so far is the ideal case.

That is,

.

we have interpreted every reflection wavelet on a seismic trace as being an actual

reflection

from a

lithological

boundary.

Actually,

many of the

"wiggles"on a trace are not true reflections, but are actually the result seismic noise.

of

Seismic noise can be grouped under two categories-

(i) Random Noise - noise which is uncorrelated from trace to trace and is •ue mainly to environmental factors.

(ii)

CoherentNoise - noise which is predictable on the seismic trace but

is unwanted. An exampleis multiple reflection interference. Randomnoise can be thought of as the additive componentn(t) which was

seen in the equationon page 2-g. Correcting for this term is the primary reason for stackingour •ata. Stackingactually uoesan excellent job of removing ranUomnoise.

Multiples, one of the major sources of coherent noise, are causedby multiple "bounces" of the seismic signal within the earth, as shownin Figure 2.16. They may be straightforward, as in multiple seafloor bouncesor "ringing", or extremelycomplex,as typified by interbed multiples. Multiples cannot be thoughtof as additive noise andmustbe modeledas a convolution with the reflecti

vi ty.

Figure 2.17

shows the

theoretical

multiple

sequence which

generatedby the simple blocky modelshown on Figure •. 5. this

data,

it

is

Multiples maybe partially

powerful

the

multiples

be

removed by

stacking,

but

important that

elimination

technique.

aleconvolution, f-k filter.ing,

Such

techniques

and inverse velocity

If

we are

effectively

would to

be

invert

removed.

often require a more include

stacking.

predictive

These techniques

wil 1 be consi alered in Part 4.

Part

2 - The Convolutional

Model

Page 2 -

18

Introduction to Seismic Inversion Methods

Fig.

2.16. Several multiple generating mechanisms.

TIME

TIME

[sec)

[sec)

0.7

REFLECTION COEFFICIENT

SERIES

Fig.

Brian Russell

2.17.

0.7

R.C.S. WITH

ALL

MULTIPLES

Reflectivi ty sequenceof Fig.

2.5.

with

and without multipl es.

.

Part 2 - The ConvolutionalModel

Page 2 -

19

PART 3 - RECURSIVE INVERSION - THEORY m•mmm•---'

.•

,-

-

-

'

•-

Part 3 - Recurstve Inversion - Theory

-

_

-

-

_-

_

Page 3 -

•ntroduct•on

to SeJsmic Znversion Methods

Brian Russell

PART 3 - RECURSIVE INVERSION - THEORY 3.1

Discrete ,

!

ß

Inversion ,

,

•

In section 2.2, we saw that reflectivity was defined in terms of acoustic impedancechanges. The formula was written:

Y•i+lV•+l ' •iV! 2i+1' Zi ri--yoi'+lVi+l+ Y•iVi---Zi..+l + Zi where

r -- refl ecti on coefficient,

/0-- density, V -- compressional velocity, Z -- acoustic impedance, and

Layer i overlies Layer i+1.

If we have the true reflectivity available to us, it is possible to recover the a.coustic impedanceby inverting the above formula. Normally, the inverse' formulation is simply written down,but here we will supply the missing steps for completness. First,

notice that:

Zi+l+Zi

Zi+1- Zt

2 Zi+1

Zi+l+Zi

Zi+1- Zi

2 Zf[

Zi+l+ Zi

Zi+l+Zi

Zi+l+Zi

I +ri- Zi+l +Zi + Zi+l +2i

Zi+l +Zi

Also

I-

ri--

Ther'efore

Zi+l Zi

l+r.

1

1

ill,

Part 3 - RecursiveInversion- Theory

ß

,

I

Page

Introduction to Seismic Invers-•onMethods

Brian Russell

pv-e-

TIME

(sec]

0.7

REFLECTION COEFFICIENT SERIES

Fig.

,

!

Part

3.1,

RECOVERED ACOUSTIC IMPEDANCE

Applying the recursiveinversionformulato a simple,and exact, reflectivity.

ß

3 - Recursive

Inversion - Theory

Page 3 -

!ntroductt •9r•

on to Se1 smJc ! nversi on Methods

;• • •;• • • •-••

9rgr•t-k'k9r9r• •-;• ;•

Or, the final

Brian

Russell

.................................................

•esult-

l+r i

Zi+[=Z

.

ß

This is called

the discrete

recursive

inversion

formula and is the basis

of many current inversion techniques. The formula tells us that if we know the acoustic impedanceof a particular layer and the reflection coefficient at the base of that layer, we may recover the acoustic impedance of the next layer. Of course we need an estimate of the first layer impedanceto start us off. Assumewe can estimate this value for layer one. Then

l+rl

Z2: Zli r1

,

Z3= Z2 11 +r2 -

To find the nth impedancefrom the first,

Figure reflection

3.1

shows the

coefficients

application

derived

in

and so on ...

r

we simply write the formula as

of the

section

recursive

2.2.

formula to

As expected,

the

the full

acoustic impedance was recovered. Problems •

ß

encountered

,

When the

m

with

i

i

•

real i

data !

m

recursive inversion formula is applied to real data,

that two serious problems are encountered.

we find

These problems are as follows-

(i) FrequencyBandlimi ti ng _

ß

Referring back to Figure 2.2 bandlimited

when

it

we see that

is convolved

with

the reflectivity

the seismic

wavelet.

is severely Both

the

low frequency componentsand the high frequency componentsare lost.

Part 3 - Recursive Inversion

- Theory

Page 3 -

4

"

Introduction to SeismicInversion Methods

0.2

0

Brian Russell

V•) 'V,•

•R

Vo:1000 m

Where: {ASSUME j•: l)

--• V,•= 1000 i-o.t

R = +0.2

-- 1500 •ec'. m (a) - 0.1

'•0.2

R• R=

Vo=1000m

= 818ii•.m R•=-0.1 -'+ ¾1 R =+0.2 R: -0.1

Figure 3.2 Effect of banUlimitingon reflectivity, single reflection coefficient,

where(a) shows

anU (b) showsbandlimited

refl ecti on coefficient. I

__

___

i

_

i

Part 3 - Recursire Inversion - Theory

i

m

i

m

I

Page 3 -

Introduction

to Seismic

(ii)

Noise

The

inclusion

Inversion

of coherent

makethe estimate• reflectivity

Methods

or random noise

let

us first

into

the seismic

Russell

'trace

will

deviate from the true reflectivity.

To get a feeling for the severity inversion,

Brian

of

the above limitations

use simple models. To illustrate

on recursire

the

effect

of

bandlimiting, consider Figure 3.Z. It shows the inversion of a single spike (Figure 3.2 (a)) anUthe inversion of this spike convolved with a Ricker wavelet (Figure 3.2 (b)). Even with this very high frequency banUwidth

wavelet, we have totally lost our abil.ity to recover the low frequency componentof the acoustic impedance. In Figure 3.3 the model derived in

minimum-phase wavelet.

section Z.2 has been convolved with a

Notice that the inversion of the data again shows a

loss of the low frequency component. The loss of the low frequency component is the most severe problem facing us in the inversion of seismic data, for is extremely

Oifficult

to

directly

recover

spectrum, we may recover muchof

the

it.

original

At

the

it

high end of the ß

frequency content using

deconvolution techniques. In part 5 we will address the problem of recovering the low frequency component. Next,

consider

sources, but will

reflectivity. reflection train

the

problem of noise.

always tend

to

interfere

This noise with

our

may be from many recovery of the true

Figure 3.4 showsthe effect of adding the full (including

multiple

transmission losses) to the model reflectivity.

As we can see on the diagram, the recovered acoustic impedancehas the basic shape as the true acoustic impedance, but becomesincreasingly

same

incorrect

with depth. This problemof accumulatingerror is compoundeU by the amplitude problemnsintroduced by the transmission losses.

Part 3 - Recurslye Inversion - Theory

Page 3 -

6

Introduction

to Seismic Invers,ion Methods

Brian Russell

pv-•,

TIME

0.?

REFLECTION COEFFICIENT SERIES

Fig.

3.3.

RECOVERED ACOUSTIC IMPEDANCE

The effect

SYNTHETIC

INVERSION

(MWNUM-PHASE WAVELET)

OF SYNTHETIC

of bandlimiting on recurslye inversion.

TIME

TIME

(see)

(re.c)

0.7

REFLECTION COEFFICIENT SERIES

Fig.

Part 3 - Recursive

3.4.

RECOVERED ACOUSTIC IMPEDANCE

The effect

Inversion

- Theory

R.C.S. WITH ALL MULTIPLES

of noise on recursive

RECOVERED ACOUSTIC IMPEDANCE

inversion.

Page 3 -

Introduction

to Seismic Inversion Methods

3.3 Continuous

Brian Russell

Inversion

A logarithmic

relationship

is

often used to

approximate the above

formulas. This is derived by noting that we can write r(t) function in the following way:

as a continuous

r(t) -- Z(t+dt) Z(t+dt)+- Z{t) _ 1 d Z(t) Z(•) - •' z'(t)

ß Or

! d In Z(t)

r(t) = •

The inverse

formula

dt is

thust

Z(t)=Z(O) exp 2y r(t)dt. 0

The precedingapproximation is valid if case.

r(t)