Russell - Introduction to Seismic Inversion Methods

Introduction

to

Seismic Inversion Methods

Brian H. Russell

Hampson-Russell

Software

Services,

Ltd.

Calgary,

Alberta

Course Notes Series, No. 2

S. N. Domenico, Series Editor

These course notes are published without the normal SEG peer reviews. They have not been examined for accuracy and clarity. Questions or comments by the reader should be referred directly to the author.

ISBN 978-0-931830-48-8 (Series) ISBN 978-0-931830-65-5 (Volume)

Library of Congress Catalog Card Number 88-62743 Society of Exploration Geophysicists

P.O. Box 702740

Tulsa, Oklahoma 74170-2740

¸ 1988 by the Society of Exploration Geophysicists

All rights reserved. This book or portions hereof may not be reproduced in any form without permission in writing from the publisher.

Reprinted 1990, 1992, 1999, 2000, 2004, 2006, 2008, 2009

]: nl;roduc t1 on •o Selsmic I nversion •thods Bri an Russell

Table of Contents

PAGE

Part I Introduction 1-2

Part Z The Convolution Model 2-1

Part 3

Part 4

Part 5

P art 6

P art 7

2.1 Tr•e Sei smic Model

2.2 The Reflection Coefficient Series

2.3 The Seismic Wavelet

2.4 The Noise Component Recursive Inversion - Theory

3.1 Discrete Inversion

3.2 Problems encountered with real

3.3 Continuous Inversion

data

Seismic Processing Consi derati ons

4. ! I ntroduc ti on

4.2 Ampl i rude recovery

4.3 Improvement of vertical

4.4 Lateral resolution 4.5 Noise attenuation

resolution

Recursive Inversion - Practice

5.1 The recursive inversion method

5.2 Information in the low frequency component

5.3 Seismically derived porosity Sparse-spike Inversi on

6.1 I ntroduc ti on

6.2 Maximum-likelihood aleconvolution and inversion

6.3 The L I norm method 6.4 Reef Problem

I nversi on appl ied to Thi n-beds

7.1 Thin bed analysis

7.Z Inversion compari son of thin beds

Model-based Inversion

B. 1 I ntroducti on .

8.2 Generalized linear inversion

8.3 Seismic 1 ithologic roodell

ing (SLIM)

Appendix 8-1 Matrix applications in geophysics Part 8 2-2 2-6 2-12 2-18 3-1 3-2 3-4 3-8 4-1 4-2 4-4 4-6 4-12 4-14 5-1 5-2 5-10 5-16 6-1 6-2 6-4 6-22 6-30 7-1 7-2 7-4 8-1 8-2 8-4 8-10 8-14

Introduction to Seismic Inversion Methods Brian Russell

Part 9 Travel-time Inversion

g. 1. I ntroducti on

9.2 Numerical examples of traveltime inversion

9.3 Seismic Tomography

Part 10 Amplitude versus offset (AVO) Inversion

10.1 AVO theory

10.2 AVO inversion by GLI Part 11 Velocity Inversion

I ntroduc ti on

Theory and Examples Part 12 Summary 9-1 9-2 9-4 9-10 10-1 10-2 10-8 11-1 11-2 11-4 12-1

Introduction to Seismic •nversion Methods Brian Russell

PART I - INTRODUCTION

Introduction to Seismic Inversion Methods Brian Russell

I NTRODUCT ION TO SE I SMI C INVERSION METHODS

, __ _• i i _ , . , , ! • _, l_ , , i.,. _

Part i - Introduction _ . .

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

the inversion of seismic data. It would therefore seem appropriate 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 measurements made on

the surface of the earth.

The above definition is so broad that it encompasses virtually all the

work that is done in seismic analysis and interpretation. Thus, in this

course we shall primarily 'restrict 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 shown in Figure 1.1.

To understand seismic inversion, we must first understand the physical

processes involved in the creation of seismic data. Initially, we will

therefore look at the basic convolutional model of the seismic trace in the

time and frequency domains,

considering the thre e components

of this model:

Part I - Introduction

_ m i --.

Introduction to Seismic InverSion Methods Brian Russell

FORWARD MODELL I NG

i m ß

INVERSE MODELLING (INVERSION) _

, ß ß _ Input' Process: Output'

EARTH

MODEL

EARTH

MODEL

Figure 1.1 Fo.•ard

' andsInverse

Model,ling

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 methods which are currently

ß

used to invert seismic data. These methods are summarized in Figure 1.2. The

primary emphasis of the course will be

the ultimate resul.t, as was previously

on poststack seismic inversion where

o

Oiscussed, is a pseudo-impeaance

section.

We will start by looking at the most contanon methods of poststack

inversion, which are based on single trace recursion. To better unUerstand

these recurslye inversion procedures, it is important to look at the

relationship between aleconvolution anU inversion, and how Uependent each

method is on the deconvolution scheme Chosen. Specifically, we will consider

classical "whitening" aleconvolution methods, wavelet extraction methods, and

the newer sparse-spike deconvolution methods such as Maximum-likelihood

deconvolution and the L-1 norm metboa.

Another important type of inversion method which will be aiscussed is

model-based inversion, where a geological moael is iteratively upUated to finU

the best fit with the seismic data. After this, traveltime inversion, or

tomography, will be discussed along with several illustrative 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 of

each seismic inversion roethoU, looking at examples of each.

Introduction to Selsmic Inversion Methods Brian Russell

SE

I SMI

C I NV

ERSI

ON

.MET•OS ,,,

MODEL-BASED

I RECURSIVE

INVERSION

• ,INVE

SION

- "NARROW

!TOMOGRAPHY)

SPARSE-

SPIKE

WAV

EF

I EL

D

NVERSIOU

I METHODS

]

Figure 1.2 A summary of current inversion techniques.

Introduction to Seismic Inversion Methods Brtan Russell

PART

2 - THE CONVOLUTIONAL

MODEL

Introduction to Seismic Inversion Methods Brian Russell Part 2 - The Convolutional Mooel

2.1 Th'e Sei smi c Model

The most basic and commonly

used one-Oimensional

moael for the seismic

trace is simply the convolution of 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 assumption

is to consiUer the 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) = w{t) * r(t).

In seismic processing

we deal exclusively with digital data, that is,

data sampled

at a constant

time interval.

If we consiUer

the 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, convolution can be

thought of as "replacing" each reflection. coefficient with a scaled version of

the wavelet and summing

the result. The result of this process

is illustrated

in Figures 2.1 and 2.Z for both a "sparse"

and a "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 spaced reflectors.

Introduction to Seismic Inversion Nethods Brian Russell WAVELET:

(a) ' * • •

: -' ':'

Convolution

of a wavelet with a

(a) •avelet. (b) Reflectivit.y.

sparse" reflectivity.

(a) (b') ! . i : !

!

:

:

Fi õure 2.2 Convolution of a wavelet with a sonic-derived "dense"

reflectivity. (a) Wavelet. (b) Reflectivity. (c) Seismic Trace

, i , ß .... ! , m i i L _ - '

Introduction to Seismic Inver'sion Methods Brian Russell

An alternate, but equivalent, way of looking at the seismic trace is in

the frequency domain. If we take the Fourier transform of the previous

ß

equati on, we may write

S(f) = W(f) x R(f),

where

S(f) = Fourier transform

of s(t),

W(f) = Fourier transform of w(t),

R(f) = Fourier transform of r(t), ana f = frequency.

In the above equation we see that convolution becomes multiplication 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) = e

w

where

(f) + er(f),

I •ndicates

amplitude

spectrum,

and

0 indicates phase spectrum. .

In other words, convolution involves multiplying the amplitude spectra

and adding the phase spectra. Figure 2.3 illustrates the convolutional model

in the frequency domain. Notice that the time Oomain problem of loss of

resolution becomes one of loss of frequency content in the frequency domain.

Both the high and low frequencies of the reflectivity have been severely

reOuceo by the effects of the seismic wavelet.

Introduction to Seismic Inversion Methods Brian Russell

AMPLITUDE SPECTRA _{PHASE SPECTRA}

w (f) I I -t- R (f) i i , I ! i. iit |11 loo s (f) I i! I i i

Figure 2.3 Convolution in the frequency domain for the time series shown in Figure 2.1.

Introduction to Seismic Inversion Methods Brian Russell

2.g The Reflection Coefficient Series

l_ _ ,m i _ _ , _ _ m_ _,• , _ _ ß _ el

of as the res within the ear

compres si onal i ropedance to re impedances by

coefficient at fo11 aws:

'The reflection coefficient series (or reflectivity, as it is also called)

described

in the previous

section

is one

of the fundamental

physical

concepts

in the seismic method. Basically, each reflection coefficient may be thought

ponse of the seismic wavelet to an acoustic impeUance change

th, where acoustic impedance is defined as the proUuct of velocity and Uensity. Mathematically, converting from acoustic

flectivity involves dividing the difference in the acoustic

the sum of the acoustic impeaances. This gives t•e reflection

the boundary between the two layers. The equation is as

•i+lVi+l - iVi

Zi+l- Z

i

i • i+1 where and r = reflection coefficient,

/o__ density,

We must also convert from depth to time by integrating the sonic log transit times. Figure •.4 shows a schematic sonic log, density log, anU resulting acoustic impedance for a simplifieU earth moael. Figure 2.$ shows

the result of converting

to the reflection

coefficient

series

and

integrating

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 nonnormal inciaence.

Introduction to Seismic Inversion Methods Brian Russell

STRATIGRAPHIC SONIC LOG

SECTION •T (•usec./mette) 4OO SHALE ... DEPTH ß ß ß ß ß ß SANOSTONE . . - .. ,

'

I

v-- I

V--3600

J

I

Fig. 2.4. Borehole

Log Measurements.

mm mm rome m .am ,mm mm m --- mm SHALE ... OEPTH

•---'-

[

Fig. 2.5. Creation of Reflectivity Sequence.

IntroductJ on 1:o Sei stoic Inversion Herhods Bri an Russell

Our best method of observing seJsm•c impedance and reflectivity is •o

derlye them from well log curves. Thus, we may create an impedance curve by

multiplying together •he sonic and density logs from a well. We may •hen

compute the reflectivlty by using •he formula shown earlier. Often, we do not

have the density log available• to us and must make do with only the sonJc. The

approxJmatJon of velocJty to •mpedance 1s a reasonable approxjmation, and

seems

to hold well for clas;cics and carbonates (not evaporltes, however).

Figure 2.6 shows the 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 assumptions which are made about the seismic

reflectivity and wavelet. Therefore, how can we describe the reflectivity seen

in a well? The traditional answer has always been that we consider the

reflectivity to be a perfectly random sequence and, from Figure •.6, this

appears to be a good assumption. A ranUom sequence has the property that its

autocorrelation is a spike at zero-lag. That is, all the components of the

autocorrelation are zero except the zero-lag value, as shown in the following

equati on-

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

)

t

zero-lag.

Let us test this idea on a theoretical random sequence, shown in Figure

2.7. Notice that the autocorrelation of 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. We see that it is even less "random" than the random spike sequence. We will discuss this in more detail on the next page.

IntroductJon to Se•.s=•c Inversion Methods Br•an Russell

RFC

F•g. 2.6. Reflectivity

sequence

derived

from

sonJc

.log.

RANDOM SPIKE SEQUENCE

WELL LOG DERIVED REFLECT1vrrY

AUTOCORRE•JATION OF RANDOM SEQUENCE

AUTOCORRELATION OF REFLECTIVITY

Fig. 2.7. Autocorrelat4ons of random and well log

der4ved

spike sequences.

Introductlon to Sei smic Inversion Methods Brian Russel 1

Therefore, the true earth reflectivity cannot be considered as being

truly random. For a typical Alberta well we see a number

of large spikes

(co•responding

to major lithol ogic change)

sticking up above

the crowd. A good

way to describe this statistically is as a Bernoulli-Gaussian

sequence. The

Bernoulli part of this term implies a sparseness in the positions of the

spikes and the Gaussian

implies a randomness

in their amplitudes. When we

generate such a sequence, there is a term, lambda, which controls the

sparseness of the spikes. For a lambda

of 0 there are no spikes, and for a

lambda

of 1, the sequence

is perfectly Gaussian in distribution.

Figure 2.8

typical Alberta well log reflectivity

would have a lambda

value in the 0.1 to

I ntroducti on to Sei smic I nversi on Methods Brian Russell It tl I I I LAMBD^•0.01 i I I •11 I 511 t •tl I (VERY SPARSE) 11 311 I LAMBDA--O. 1 4# I 511 I #1 I TZIIE (KS ! 1,1

::.

•"• •'•;'"

' "";'•'l•'

"••'r'•

- • "(11

I

EXAMPLES

OF REFLECTIVITIES

Fig. 2.8. Examples of reflectivities using lambda

factor to be discussed in Part 6.

, , m i ß i

Introduction to Seismic Inversion ,Methods Brian Russell

2.3 The Seismic Wavelet

-- _ ß • ,

Zero Phase and Constant Phase Wavelets

m _ m _ m ß m u , L m _ J

The assumption tha.t there is a single, well-defined wavelet which is convolved with the reflectivity to produce the seismic trace is overly simplistic. More realistically, the wavelet is both time-varying and complex in shape. However, the assumption of 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 dependent only on its dominant frequency, that is, the peak frequency of its a•litude spectrum or the inverse of the dominant period in the time domain (the dominant period is

found by measuring

the time from trough to trough). Two Ricker wave'lets are

shown in Figures 2.9 and 2.10 of frequencies 20 and 40 Hz. Notice that as the anq•litude spectrum of a wavelet .is broadened, the wavelet gets narrower in the

time domain,

indicating

an increase

of resolution. Our ultimate

wavelet

would

be a spike, with a flat amplitude spectrum. Such a wavelet is an unrealistic goal in seismic processing, but one that is aimed for.

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

same wavelet has been shifted by 30 degree increments. Notice that the 90

degree rotation displays perfect antis•nmnetry, whereas a 180 degree shift

simply inverts the wavelet. The 30 degree rotations are asymetric.

Introduction to Seismic Inversion Methods Brian Russell

Fig.

Fig.

2.9. 20 Hz Ricker Wavelet'.

•.10. 40 Hz Ricker wavelet.

Fig. 2.11. Ricker wavelet rotated by 90 degree increments

Fig.

Part 2 - The Convolutional Model

2.12. Ricker wavelet rotated by 30 degree increments

Introduction to Seismic Inversion Methods Brian Russell

Of course, a typical seismic wavelet contains a larger range of

frequencies than that shown on the Ricker wavelet. Consider the banapass

fil•er shown

in Figure 2.13, where we have passed a bana of frequencies

between 15 and 60 Hz. The filter has also had cosine tapers applied between 5

and 15 Hz, and between 60 and 80 Hz. The taper reduces the "ringing" effect

that would be noticeable if the wavelet amplitude spectrum was a simple

box-car. The wavelet of Figure 2.13 is zero-phase, and would be excellent as

a stratigraphic wavelet. It is often referred to as an Ormsby wavelet.

Minimum Phase Wavelets

The concept of minimum-phase is one that is vital to aleconvolution, but

is also a concept that is poorly understood. The reason for this lack of

understanding is that most discussions of the concept stress the mathematics

at the expense of the physical interpretation. The definition we

use of minimum-phase is adapted from Treitel and Robinson (1966):

For a given set of wavelets, all with the same amplitude spectrum,

the minimum-phase

wavelet

is the one

which

has the sharpest

leading

The reason that 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 is also minimum-phase. The minimum-phase

equivalent of the 5/15-60/80 zero-phase wavelet is shown in Figure 2.14. As

in the aefinition used, notice that the minimum-phase wavelet has no component

prior to time zero and has its energy concentrated as close to the origin as

possible. The phase spectrum of the minimum-wavelet is also shown.

I•troduct•on to Sei stoic !nversion

Nethods.

Br•an Russell

ql Re• R Zero Phase I•auel•t 5/15-68Y88 {•

0.6

f1.38 - Trace 1

iii

- e.3e

... ,

• ...

'

2be

1 Trace I

Fig. 2.13. Zero-phase bandpass

wavelet.

Reg 1) min,l• wavelet •/15-68/88 hz

18.00 p Trace I

Reg E wayel Speetnm

'188.88

•

Trace 1

0.8

188

Fig. 2.14. Minim•-phase equivalent

of zero-phase wavelet

shown in Fig. 2.13.

_

! m,m, i m

Part 2 -Th 'e Convolutional Model

i

Introduction to Seismic Inversion Methods Brian Russell

Let us now look at the effect of different wavelets on the reflectivity

function itself. Figure 2.15 a anU b shows a number of different wavelets

conv6lved with the reflectivity (Trace 1) from the simple blocky model shown

in Figure Z.5. The following wavelets have been used- high

zero-phase (Trace •),

low frequency

zero-phase

(Trace ½), high

minimum phase (Trace 3), low frequency minimum phase (Trace 5).

figure, we can make the fol 1 owing observations:

frequency frequency

From the

(1) Low freq. zero-phase wavelet: (Trace 4)

- Resolution of reflections is poor.

- Identification of onset of reflection is good.

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

- Resolution of reflections is good.

- Identification of onset of reflection is good.

(3) Low freq. 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 above observations, we would have to consider the high

frequency,

zero-phase

wavelet

the best, and the low-frequency,

minimum

phase

wavelet the worst.

(a)

Introduction to Seismic Inversion Methods Brian Russell

!ql Reg R Zer• Phase Ua•elet •,'1G-•1• 14z

F

- •.• ['

'

•,3 Recj B miniilium phue ' '

17 .•

q2 Reg C Zero Phase 14aue16(' ' •'le-3•4B Hz

e

q• Reg 1) 'minimum phase " •,leJ3e/4e h• '

8

e.e •/••/'•-•"v--,._,,

-r

e.•

' ' " s•e

''

2.15. Convolution of four different wavelets shown in (a) with trace I of (b). The results are shown on traces 2 to 5 of (b).

Introduction to Seismic Inversion Methods Brian Russell

g.4 Th•N.

oi se. C

o.

mp.o•ne

nt

-

The 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 of

(i) Random Noise - noise which is uncorrelated from trace to trace and is

•ue mainly to environmental factors.

(ii) Coherent Noise - noise which is predictable on the seismic trace but is unwanted. An example is multiple reflection interference.

Random noise can be thought of as the additive component n(t) which was

seen in the equation on page 2-g.

Correcting for this term is the primary

reason for stacking our •ata.

Stacking actually uoes an excellent job of

Multiples, one of the major sources of coherent noise, are caused

by

multiple "bounces"

of the seismic signal within the earth, as shown

in Figure

2.16. They may be straightforward, as in multiple seafloor bounces

or

"ringing", or extremely

complex, as typified by interbed multiples. Multiples

cannot be thought of as additive noise and must be modeled

as a convolution

Figure

generated by the simple blocky model

this data, it is important that

Multiples may be partially removed

powerful elimination technique.

aleconvolution, f-k filter.ing, wil 1 be consi alered in Part 4.

2.17

shown on Figure •. 5. the multiples be

by stacking, but

Such techniques and inverse velocity stacking.

shows the theoretical multiple sequence which would be

If we are to invert

effectively removed. often require a more include predictive

These techniques

Introduction to Seismic Inversion Methods Brian Russell

Fig. 2.16. Several multiple generating mechanisms.

TIME TIME

[sec) [sec)

0.7 0.7

REFLECTION R.C.S.

COEFFICIENT WITH ALL

SERIES MULTIPLES

Fig. 2.17.

_{Refl ectivi ty sequence}

_{of Fig.}

and without mul tipl es.

Part 2 - The Convolutional Model

2.5. with

.

PART 3 - RECURS IVE INVERSION - THEORY

m•mmm•---' .• ,- - - ' •- - _ - - _- _

•ntroduct•on to SeJsmic Znversion Methods Brian Russell

PART 3 - RECURSIVE INVERSION - THEORY

3.1 Discrete Inversion

, ! ß , , •

In section 2.2, we saw that reflectivity was defined in terms of

acoustic impedance changes. The formula was written:

Y•i+lV•+l

' •iV! 2i+

1' Z

i

ri-- yoi'+lVi+l+

Y•iVi

-- -Zi..+l

+ Z

i

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 impedance by 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:

Also

Ther'efore

Zi+l+

Z

i

Zi+

1- Z

t

2 Zi+

1

I + ri- Zi+l

+ Zi + Zi+l

+ 2i Zi+l

+ Zi

I- ri--

Zi+l+

_{Zi+l+ Z i}

Z

i

Zi+

_{Zi+l+ Z i}

1- Z

i

_{Zi+l+ Z i}

2 Zf[

Zi+l

Z i

l+r.

1

1

Part 3 - Recursive Inversion- Theory

ill, ß , I

Introduction to Seismic Invers-•on Methods Brian Russell pv-e- TIME (sec] 0.7 REFLECTION COEFFICIENT SERIES RECOVERED ACOUSTIC IMPEDANCE

Fig. 3.1,

Applying

the recursive

inversion

formula

to a

simple, and exact, reflectivity.

, ! ß

!ntroductt on to Se1 smJc ! nversi on Methods Brian Russell •9r• ;• • •;• • • •-•• 9rgr•t-k'k9r9r• •-;• ;• ...

Or, the final •esult-

Zi+[= Z

ß

l+r i

.

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 impedance of 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 impedance to start us

off. Assume we can estimate this value for layer one. Then

l+rl

,

Z2:

Zl i r 1

Z3=

Z

2 11

+ r 2

and so on ...

To find the nth impedance from the first, we simply write the formula as

Figure 3.1 shows the application of the recursive formula to the "

reflection coefficients derived in section 2.2. As expected, the full acoustic impedance was recovered.

Problems encountered with real data

• ß , m i i • i ! m

When the recursive inversion formula is applied to real data, we find

that two serious problems are encountered. These problems are as follows-

(i) Frequency Bandl imi ti ng _ ß

Referring back to Figure 2.2 we see that the reflectivity is severely

bandlimited when it is convolved with the seismic wavelet. Both the

low frequency components and the high frequency components are lost.

Introduction to Seismic Inversion Methods Brian Russell

0.2

0

V•) 'V,•

V

o: 1000

m

Where:

--• V,• = 1000 i-o.t

R•

R=

{ASSUME

j•: l)

R•=

-0.1

V

o= 1000 m

-'+ ¾1

= 818 m

Figure 3.2 Effect of banUlimiting on reflectivity, where (a) shows

single reflection coefficient, anU (b) shows bandlimited

refl ecti on coefficient.

i i m i m I

I __ ___ i _

Introduction to Seismic Inversion Methods Brian Russell

(ii) Noise

The inclusion of coherent or random noise into the seismic 'trace will

make the estimate• reflectivity deviate from the true reflectivity.

To get a feeling for the severity of the above limitations on recursire

inversion, let us first use simple models. To illustrate the effect of

bandlimiting, consider Figure 3.Z. It shows the inversion of a single spike (Figure 3.2 (a)) anU the 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

component of the acoustic impedance.

In Figure 3.3 the model derived in section Z.2 has been convolved with a

minimum-phase wavelet. 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 it

is extremely Oifficult to directly recover it. At the high end of the ß

spectrum, we may recover much of the original frequency content using

deconvolution techniques. In part 5 we will address the problem of recovering

the low frequency component.

Next, consider the problem of noise. This noise may be from many sources, but will always tend to interfere with our recovery of the true

reflectivity.

Figure 3.4 shows the effect of adding the full multiple

reflection train (including transmission losses) to the model reflectivity.

As we can see on the diagram, the recovered acoustic impedance has the same

basic shape as the true acoustic impedance, but becomes increasingly incorrect

with depth. This problem of accumulating error is compoundeU by the amplitude

problemns introduced by the transmission losses.

Introduction to Seismic Invers,ion Methods Brian Russell TIME Fig. TIME (see) Fig. 0.? RECOVERED ACOUSTIC IMPEDANCE REFLECTION SYNTHETIC COEFFICIENT (MWNUM-PHASE SERIES WAVELET) pv-•, INVERSION OF SYNTHETIC

3.3. The effect of bandlimiting on recurslye inversion.

0.7

TIME

(re.c)

REFLECTION RECOVERED R.C.S. RECOVERED

COEFFICIENT ACOUSTIC WITH ALL ACOUSTIC

SERIES IMPEDANCE MULTIPLES IMPEDANCE

3.4. The effect of noise on recursive inversion.

Introduction to Seismic Inversion Methods Brian Russell

3.3 Continuous Inversion

A logarithmic relationship is often used to approximate the above formulas. This is derived by noting that we can write r(t) as a continuous function in the following way:

Or

r(t) - Z(t+dt)

- Z{t) _ 1 d Z(t)

ß - Z(t+dt) + Z(•) - •' z'(t)

! d In Z(t)

r(t) = •

dt

The inverse formula is thus-

t

Z(t)

= Z(O)

exp

2y r(t) dt.

0

The preceding

approximation

is valid if r(t) <10.3• which is usually the

case. A paper by Berteussen and Ursin (1983), goes into much more detail on

the continuous versus discrete approximation. Figures 3.5 and 3.6 from their

paper show that the accuracy of the continuous inversion algorithm is within

4% of the correct value between reflection coefficients of -0.5 and +0.3.

If our reflection coefficients are in the order of + or - 0.1, an even

simpler

approximation

may

be made

by dropp'ing

the logarithmic

relationship:

t

1 d Z(t)

•_==•

Z(t)

--2'Z(O)

fr(t) dt

r(t)

--• -dr

VO

Introduction to Seismic Inversion Methods Brian Russell Fig. 3.5 m i ,, ,m I I IIIII

I + gt

½xp

(26•)

Difference

Numerical c•pari son of discrete and continuous i nversi on.

(Berteussen and Ursin, 1983)

Fig. 3.6

$000

} m

MPEDANCE

(O

I SCR.

)

r-niL

${300

-•

O

I FFERENCE

SO0 O I FFERENCE ( SCALED UP )

T •'•E t SECONOS

C•pari son between

impedance

c•putatins based

on a

discrete and a continuous

seismic •del.

(Berteussen and Ursin, 1983)

Introduction'to Seismic Inversion Methods Brian Russell

PART 4 - SEISMIC PROCESSING CONSIDERATIONS

•ntroduction to Seismic •nvers•on Methods B.r. ian Russell

4.1 Introduction

Having looked at a simple model'of the seismic trace, anu at the

recursire inversion alogorithm in theory, we will now

look at the problem of

processing

real seismic eata in order to get the best results from seismic

inversion.

We may group the key processing

problems into the following

( i ) Amp 1 i tu de rec o very.

(i i) Vertical resolution improvement.

(i i i ) Horizontal resol uti on improvement.

(iv) Noise elimination.

Amplitude problems are a major

consideration

at the early processing

stages

and we will look at both deterministic amplitude

recovery

and surface

consistent residual static time corrections. Vertical resolution improvement

will involve a discussion of aleconvolution and wavelet processing techniques. In our discussion of horizontal resolution we will look at the resolution

improvement

obtained in migration, using a 3-D example. Finally, we will

consider several approaches

to noise elimination, especially the elimination

of multi pl es.

Simply stateu, to invert our

one-dimensional model given in the

approximation of this model (that

band-limited reflectivity function)

these considerations in minU. Figure 4.1

be useU to do preinversion processing.

seismic data we usually assume the

previous section. And to arrive at an

is, that each trace is a vertical,

we must carefully process our data with shows a processing flow which could

Introduction to Seismic Inversion Methods Brian Russell

INPUT RAW DATA

DETERMINISTIC AMPLITUDE CORRECTIONS ,. _•m mlm SURFACE-CONS ISTENT

DECONVOLUTIO,

N FOLLOWED

VELOCITY ANAUYS IS

APPbY STATICS AND VEUOCITY

MULTIPLE ATTENUATION

STACK

ß •

MI GRATI ON

,

Fig. 4.1.

Simpl

i fied i nversi

on

processing

flow.

ll , ß ' ß I , _ i 11 , m - -- m _ • • ,11

Inl;roducl:ion 1:o SeJ smlc Invers1 on Nethods BrJ an Russell

4.2 Am.p'l i tu. de.. P,.ecovery

The most dJffJcult job in the p•ocessing of any seismic line is

•econst•ucting

the amplJtudes

of the selsmJc

t•aces as they would

have been

Jf

the•e

were no dJs[urbJng inf'luences present. We normally make the

simplJfication

that the distortion of the seJsmic

amplJtudes

may

be put into

three main categories' sphe•Jcal

divergence, absorptJon, and t•ansmJssion

loss. Based on a consideration of these three factors, we may wrJte aown an

approximate

functJon for the total earth attenuation-

Thus, data, the

formula.

At: AO*

(

where t = time,

A

t = recorded

amplitude,

A

0 = true ampl

i tude,

anU

a,b = constants.

if we estimate the constants in the above equation from the seismic

true amplitudes

of the data coulU be recovered by using the inverse

The deterministic amplitude correction and trace to trace mean

scaling will account

for the overall gross changes in amplitude. However,

there may still

be subtle (or even not-so-subtle) amplitude problems

associated

with poor surface conditions or other factors. To compensate

for

these effects, it is often advisable to compute and apply surface-consistent

gain corrections. This correction involves computing

a total gain value for

each trace and then decomposing this single value in the four components

Aij=

Six

Rj

x G

k x MkX

•j,

where A = Total amplitude factor,

S = Shot component, R: Receiver component, G = CDP component, and

M = Offset component, X = Offset distance,

i,j = shot,receiver pos., k = CDP position.

Introduction to Seismic .Inversion Methods Brian Russell

SURFACE

CONS

Ib'TEh[O{

Fig. 4.2. _{Surface and sub-surface geometry and}

surface-consistent decomposition. (Mike Graul).

, ,

Introduction to Seismic Inversion Methods Brian Russell

Figure 4.g (from Mike Graul's unpublished course notes) shows the

geometry

used for this analysis. Notice that the surface-consistent

statics

anti aleconvolution

problem

are similar. For the statics problem, the averaging

can be •1one by straight summation. For the amplitude problem we must

transform the above equation into additive form using the logarithm:

In Aij=

In S

i + In Rj

+ In G

k + lnkMijX•.

The problem can then be treated exactly the same way as in the statics

case. Figure 4.3, from Taner anti Koehler (1981), shows

the effect of doing

4.3 I•mp.

rov.

ement_

o.[_Ver.

t.i.ca.1..Resoluti

on

Deconvol ution is a process by which an attempt is made to remove the

seismic wavelet from the seismic trace, leaving an estimate of reflectivity.

Let us first discuss the "convolution" part of "deconvolution" starting with

In the

st-- wt* r t

where

frequency domain

st = the sei smic

trace,

wt= the seismic

wavelet,

rt= reflection coefficient series,

* = convol ution operation.

S(f) • W(f) x R(f) .

The deconvol ution

procedure and consists

reflection coefficients.

fol 1 owl ng equati on-

rt: st* o

process is simply the reverse of the convolution

of "removing" the wavelet shape to reveal the

We must design an operator to do this, as in the

where Or--

operator

-- inverse

of w

t .

Part 4 - Seismic Processing Considerations

Introduction to Seismic Inversion Methods Brian Russell ii 11 ß 1' i ii '..,•' •, ," " " ß d.

Preliminary stack bet'ore surface consistent static and ompli- lude corrections.

ß Stock with surface consistent static and amplitude cor- rections.

Fig. 4.3. Stacks with and without surface-consi stent

corrections. (Taner anu Koehler, 1981).

Part 4 - Seismic Processing Considerations

ß ,

Introduction to Seismic Inversion Methods Brian Russell

In the frequency domain, this becomes R(f) = W(f) x 1/W(f) .

After this extremely simple introduction, it may appear that the

deconvolution

problem should be easy to solve. This is not the case, and the

continuing research into the problem testifies

to this.

There are two main

problems. Is our convolutional

model

correct, and, if the model is correct,

question is that the convolutional

model

appears

to be the best model

we have

come

up with so far. The main problem is in assuming

that the wavelet does

not vary with time. In our discussion we will assume that the time varying

problem

is negligible within the zone of interest.

The second

problem is much more severe, since it requires solving the

ambiguous

problem

of separating a wavelet and reflectivity sequence

when only

the seismic trace is known. To get around this problem, all deconvolution or

wavelet estimation programs

make certain restrictive assumptions,

either about

the wavelet or the reflectivity. There are two classes of deconvolution

methods: those which make restrictive phase assumptions and can be considered ,

true wavelet processing techniques only when these phase assumptions are met,

and those which do not make restrictive phase assumptions and can be

considered as true wavelet processing methods. In the first category are (1) Spiking deconvolution,

(2) Predictive deconvolution,

(3) Zero phase deconvoluti on, and

(4) Surface-consi stent deconvoluti on.

Introduction to Seismic Inversion Methods Brian Russell

(a)

Fig. 4.4 A comparison of non surface-consistent and surface-consistent

decon on pre-stack data. {a) Zero-phase deconvolution.

{b) Surface-consistent soikinB d•convolution.

(b),

Fig. 4.5 Surface-consistent decon comparison after stack. (a) Zero-phase aleconvolution. (b) Surface-consistent

deconvol ution.

'--'- , ß , ,• ,t ß ß _ , , _ _ ,, , ,_ , ,

Introduction to Seismic Invers. ion Methods Brian Russell

In the second category are found

(1) Wavelet estimation using a well

(Hampson and Galbraith 1981)

1 og (Strat Decon).

(2) Maximum-1 ikel ihood aleconvolution. (Chi et al, lg84)

Let us surface-consi stent surface-consi stent

components. We

di recti ons- common

illustrate the effectiveness of one of. the methods, aleconvolution. Referring to Figure 4.•, notice that a

scheme involves the convolutional proauct of four

must therefore average over four different geometry

source, common receiver, common depth point (CDP), and

con, non offset (COS). The averaging must be performed iteratively and there

are several different ways to perform it. The example in Figures 4.4 ana 4.5

shows an actual surface-consi stent case study which was aone in the following

way'

(a) Compute the autocorrelations of each trace,

(b) average the autocorrelations in each geometry eirection to get four

average autocorrel ati OhS,

(c) derive and apply the minimum-phase inverse of each waveform, and

(•) iterate through this procedure to get an optimum result.

Two points to note when you are looking at the case study are the

consistent definition of the waveform

in the surface-consistent

approach an•

the subsequent improvement of the stratigraphic interpretability of the stack.

We can compare all of the above techniques using Table 4-1 on the next

page. The two major facets of the techniques which will be compared are the

wavelet estimation procedure and the wavelet shaping procedure.

Introduction to Seismic Inversion Methods Brian Russell

Table 4-1 _{Comparison of Deconvol ution MethoUs}

m m ß ß m METHOD Spiking Deconvol ution Predi cti ve Deconvol uti on Zero Phase Deconvol utton Surface-cons. Deconvolution Stratigraphic Deconvol ution Maximum- L ik el i hood deconvol ution WAVELET ESTIMATION

Min.imum phase assumption

Random refl ecti vi ty

assumptions.

No assumptions about

wavelet•

Zero phase assumption.

Random refl ectt vi ty

assumption.

Minimum or zero phase.

Random reflecti vi ty

assumption.

No phase assumption.

However, well must match

sei smi c.

No phase assumption.

Sparse-spike assumption.

WAVELET SHAPING

Ideally shaped

to spike.

In practice, shaped

to minimum

phase, higher frequency

output.

Does not whiten data well.

Removes

short and long period

multiples. Does not affect

phase

of wayel

et for long lags.

Phase is not altered.

Amplitude spectrum i$

whi tened.

Can shape

to desired output.

Phase

character i s improved.

Ampl i rude spectrum i s

whitened

less than in single

trace methods.

Phase of wavelet is zeroed.

Amplitude

spectrum

not

whi tened.

Phase of wavelet is zeroed•

Amp 1 i rude spectrum i s

whi tened.