seismic data analysis

8/19/2019 Seismic Data Analysis

http://slidepdf.com/reader/full/seismic-data-analysis 1/17

[ntroductJon to Seismic Inversion Methods Br•an Russel•

PART 8 - MODEL-BASED NVERSION

_ - _ - m m L ß .... •

Part 8 - Model-based Inversion

Page 8 -



Introduction to Seismic Inversion Methods Brian Russell

8.1 Introducti on

In the past sections of the course, we have derived reflectivi-ty

information directly from the seismic section and used recursire inversion to

produce a final velocity versus depth model. We have also seen that these

methods can be severely affected by noise, poor amplitude recovery, and the

band-limited nature of seismic data. That is, any problems in the data itsel f

will be included in the final inversion result.

In this chapter, we shall consider the case of builaing a geologic moUel

first and comparing the model to our seismic data. We shall then use the

results of t•is comparison between real and modeled data to iteratively update

the model in such a way as to better match the seismic data. The basic idea

of this approac• is shown in Figure 8.1. Notice that this method is

intuitively very appealing since it avoids the airect inversion of the seismic

data itself. On the other hand, it may be possible to come up with a model

that matches he data'very well, but is incorrect. (This can be seen easily

by noting that there are infinitely manyvelocity/depth pairs that will result

in the same ime value.) This is referred to as the problem of nonuniqueness.

To implement the approach shown in Figure 8.1, we need to answer two

fundamental questions. First, what is the mathematical relationship between

the model data and the seismic data? Second, how do'we update the' model? We

shall consider two approacheso theseproblems, he generalized inear

inversion (GLI) approach outlined in CooRe and Schneider (1983}, and the

Seismic Lithologic (SLIM) method which was developed in Gelland and Larner

(1983).


Page 8 - 2



Introduction to Seismic Inversion Methods' Brian Russell

CALCULATE

ERROR

UPDATE

IMPEDANCE

ERROR

SMALL

ENOUGH

NO

YES

SOLUTION

= ESTIMATE

Model Based Invemion

Figure 8.1

Flowchart for the model based inversion technique.


Page 8 -



Introduction %o Seismic Inversion Methods Brian Russell

8.2 Generalized Linear Inversion

The generalized linear inversion(GLI) method is a methodw•ich can be.

applied to virtually any set of geophysicalmeasurementso determine the

geological situation whichproduced these results. That is, given a set of

geophysicalobservations, the GLI method ill derive the geological model

which best fits t•ese observations in a least squares sense. Mathematically,

if we express the model and observations as vectors

M: (m,m, ..... , mk)=vectorfkmodelarameters,nd

T: (t1, t2, ..... , tnT

vector of n observations.

Then the relationship between the model

in the functional form

and observations can be expressed

t i = F(ml,m , ...... , m )

ß i : 1, ... , n.

functional relationship has been derived between the

nce the

observations and the model, any set of model parameters will produce an

ß

output. But what model?GLI eliminates he need or trial and error by

analyzing the error between he model output and the observations, and then

in such a way as to produce an output which

way, we may iterate towards a solution.

perturbing the model parameters

will produce ess error. In this

Mathematically'

)F(MO)

= F(Mo) aT •M,

MO--nitial odel,

M: true earth model,

AM: change n model parameters,

F(M) : observations,

F(Mo): alculatedaluesrom nitial

•)F(M )

.2 • = changen calculatedalues.

model, and

F(M)

where


Page 8 - 4



Introduction to Seismic Inversion Flethods Brian Russell

IMPEDANCE

4.6 41.5AMPLITUDE

ml

I

ß

,

- ii

,i i,

i

i

ii

,

ß

ß

,

, i

:.

__

IMPEDANCE

(GM/CM3) FT/SEC) 1000

41.5 4.6 41.5 4.6

i i

41.5

b c d e

Figure 8.2

A synthetic test of the GLI approach to model based

inversion.

(a) Input impedance. (b) Reflectivity derived from (a)

with added multiples. (c) Recurslye inversion of (b).

(d) Recurslye inversion of (b)convolved with wavelet.

(e) GLI inversion of (b). (Cooke and Schneider, 1983)

Part 8 - Model-based I nversi on

Page 8 -




But note that the error between the observations and the computed values

i s simply

•F = F(M) F(MO).

Therefore, the above equation can be re expressed as a matrix equation

•F = A AM,

where A: matrix of deri vatives

with n rows anU k columns.

The soluti on to the above equation would appear to be

-1

•M = A •F,

where A l: matrix nverse f A.

However, since there are usually more observations than parameters (that

is, n is usually greater than k) the matrix A is usually not square and

therefore does not have a true inverse. This is referred to as an

overdeterminedcase. To solve the equation in that case, we use a least

squares solution often referred to as the Marquart-Levenburgmethod see Lines

and Treitel (1984)). The solution is given by

•M: (AT'A)-IAZ•F.

Figure 8.1 can be thought of as a flowchart of the GLI method f we make

the impedanceupdate using the method ust described. However, we still must

derive the functional relationship necessary to relate the model to the

observations. The simplest solution which presents itself is the standarO

convol utional model

s(t) = w(t) * r(t), where r(t) = primaries only.

Cooke and Schneider (1983) use a modi ied version of the previous formula

in which multiples and transmission losses are modelled. Figure 8.2 is a

composite from their paper showing he results of an inversion applied to a

single synthetic impedance trace.


Page 8 - 6




• ' • IMP.EDANCE1OOO

(GM/CM3)(FT/$EC)

ß ._ . .:. . . .........•::...., .. ... .. :... O

.o . ß ß ,, ,, ? "e'. ,,

. .:-: . .• ..... : :........:..:.-.-_- ........ , ß ....-. -.

4': ::.•/-.:. i i..::..':...:......:.':i•i.'-'-:....'...'......-...•.•.::

..'." .

• ' 300M$

.

,

Figure 8.3

2-D model to test GLI algorithm. The well on the right

encounters gas sand while the well on the left does not.

(Cooke and Schneider, 1983)

Figure 8.4

AMPLITUDE

Model traces derived from

m)del in Figure 8.3.

{Cooke and Sc)•neider, 1983)

Part 8 - Model-based •nversion

Figure 8.5

IMPEDANCE

(GM/CM3 (FT/SEC)X1000

10 38 10 38

,,,.l A B

GLI inversion of model traces. Compa

with sonic log on right side of Fi•iure

(Cooke and Schneider, 1983

Page 8 - 7



Introduction to Seismic Inver. sJon Methods Brian Russell

In Figure 8.2, notice that the advantage of incorporating multiples in

the solution is that, although they are modelled in co•uting the seismic

response, hey are not included in the model parameters. This is a big

advantage over recursire methods, since those methods incorporate the

multiples into the solution if they are not removed rom the section.

Another important feature of this particular method is the

parameterizationused. Instead of assigninga different value of velocity at

each time sample, large geological blocks were defined. Each block was

assigneda starting impedance alue, impedancegradient, and a thickness in

time. This reduceU the numberof parameters and therefore simplified the

computation.However,here is enough flexibility in this modellingapproach

to derive a fairly detailed geological inversion. We will now look at both a

syntheticandreal exampleromCooke ndSchneider1983).

A 2-0 synthetic example was next considered by Cooke and Schneider

(1983). Figure 8.3 shows he model, which consisted of two gas sands encased

in shale. One well encountered the sand and the other missed. The impedance

profile of the discovery well is shown n the right. Figure 8.4 shows

synthetic traces over the two wells, in which a noise component as been

added. Finally, Figure 8.5 shows he initial guess and the final solution,

for which the gradients have been set to zero. Notice that although the

solution is not perfect, the gas sand has been delineated.


- 8




I

I

YES ___•__•J

' ' FINALMObE

- _ ._ x•, • .... r -• •;•,• -.-'%•..

-cx-r. . . . .-. .,'•_;'•.:.

,• . . t .•..

Figure 8.6

I11 ustrated flow chart for the SLIM method.

(Western GeophysicalBrochure)


Page 8 - 9



ntroducti on to Sei stoic nver si on Methods Brian Russel 1

8.3 Sei_smic_ithologicModelling ,SLIM)

Although the n•thod outlined in Cookeand Schneider (1983) showed much

promise, it has not, as far as this author is aware, been implemented

commercially. However, one method that appears very similar and is

commercially available is the Seismic Lithologic Modeling (SLIM) method of

Western Geophysical. Although the details of the algorithm have not been

fully released, the method does involve the perturbation of a model rather

than the direct inversion of a seismic section.

Figure 8.6 shows a flowchart of the SLIM method taken from a Western

brochure. Notice that, as in the GLI method, an initial geological model is

created and comparedwith a seismic section. The model is defined as a series

of layers of variable velocity, density, and thickness at various control

points along the line. Also, the seismic wavelet is either supplied (from a

previous wavelet extraction procedure) or is estimated from the data. The

synthetic model is then comparedwith the seismic data and the least-squared

error sum is computed. The model is perturbed in such a way as to reduce the

error, and the process is repeated until convergence.

The user has total control over the constraints and may incorporate

geological information from any source. The major advantage of this method

over classical recurslye methods is that noise in the seismic section is not

incorporated. However, s in the GLI method, •hesolution is nonunique.

The best examples of applying this method to real data are given in

Gelland and Larner (1983). Figure 8.7 is taken from their paper and shows an

initial Denver basin model which has 73 flat layers derived from the major

boundaries of a sonic log. Beside this is the actual stacked data to be

inverted.

Part 8 - Model-based I nversi on

Page 8 - 10




1.4

1.6

1kit

,1.4

2.0

Initial

Figure 8.7

lkft

Stack

Left' Init)al Denver asinmodel eismic.

Right: Stacked section from DenverBasin.

(Gel and and Larner, 1983).

2.0

.4

1.6

1.8

1.8

2.0 •

Fieldata Synthetic Reflectivity 2.0

Figure 8.8 Left: F•na• SLIM JnversJon of data shown 1n

Figure 8.7 spl iceU into field data.

Right- Final reflectivity from inversion.

' -- _• -- __-__i m - ' -' (Gelfand and Larner, 1983). • .......... .m:

Part 8 - Model based Inversion Page 8 - 11




In Figure 8.8 the stack is again shown in its most complexregion, with

the final synthetic data is shown fter 7 iterations through the program.

Notice the excellent agreement. On the right hand si•e of Figure 8-.9 is the

final reflectivity section from which the pseudo mpedance s derived. Since

this reflectivity is "spi•y", or broad band, it already contains the low

frequencycomponentecessaryor full inversion. Finally, Figure 8.10 shows

the final inversion compared ith a traditional recursire inversion. Note the

'blocky' nature of the parameter ased nversion when comparedwith the

recurs i ve i nvers i on.

I n summary, parameter

which can be thought of

reflectivity is extracted.

propagated hrough the final

based inversion i s an iterative model1 ng scheme

as a geology-baseddeconvolution since the full

I• has the advantage that errors are not

result as in recursire inversion.

Part 8 - Model-based

Inversion

Page 8 - 12




w

1

500-ft 114mile 114mile E

lkft •

-

.5

l m

ß

.7

1.9

Figure 8.9 Impedance section derived from SLIM inversion of

Denver Basln 1 ine shown n Figure 8.7.

{GelfanU and Larner, 1983)

W

1.7

50011 114mileS 114mile

lkft ß ß .• E

19

F gu e 8.10

Traditional recursire inversion of Denver Basin line

from F gur. 8.7.

(Gelfana anU Larner, 1983)


Page 8 - 13




Appendix- Mat_r_ixappljc.•at.ons_inGe•ophy.s.ics

Matrix theory showsup in every aspect of geophysicalproocessing.Before

looking at generalized matrix theory, let us consider he application of

matrices to the solution of a linear equation, probably the most important

application. For example, let

3x1+2x : 1, and

x1- x2 = 2.

By inspection, we see that the solution is

However,we Could .haveexpressed the equations in the matrix form

or

A X = y,

3 2 x1

1 -1 x2 ß

The sol ution is, therefore

or

-1

x = A y,

x1 1 . -2 1

-1/5

1 3 1

x2


Page 8 - 14



Introduction to Seismic Inversion Me.thods Brian Russell

is of little

overde termi neU

problems:

In the above equations we had the same numberof equations as unknowns

and the problem t•erefore had a unique answer. In matrix terms, this means

that the problem can be set up as a square matrix of dimension N x N times a

vector of dimension N. However, in geophysical problemswe are Uealing with

the real earth anU the equations are never as nice. Generally, we either have

fewer equations than unknowns (in which case the situation is called

underdetermined) or more equations than unknownsin which case the situation

is calleU overdetermined). In geophysicalproblems, he underUeterminedase

interest to us since there is no unique solution. The

case is of much nterest since it occurs in the following

( ) Surface consi stent resi dual

(2) Lithological modelling,and

(3) Refracti on model ng.

statics,

The overdetermined system of equations • can

categories- consistent an• inconsistent. These

extending our earlier example.

be split into two separate

are best described by

(a) Con•s.i••t Overd..eterminedn.earEqua.t.on.s

In this case we

equations are simply

reUunUant equations may

square matrix case.

earl ier example,

have more equations than unknowns, but the extra

scaled versions of t•e others. In this case, the

simply be eliminated, reducing the prø•lem o the

For examp.le, consider adding a third equation to our

so that

anU

3x1+ x : 1,

x1- x2 : 2,

5x - 5x : 10.


P age 8 -

15




This may be written in matrix form as

2 x1

x

o

But notice that the third equation is simply five times the second, and

therefore conveys no new information. We may thus reduce the system of

equations back to the original form.

(b) Inco,s,s, en•tO•verd.•ermine.L.i.near qua•i.on?

In this case the extra equations are not scalea versions of other

equations-in the set, but conveyconflicting information. In this case, there

is no solution to the problem which will solve all the equations. This is

usually the case in our seismic wor• and indicates the presence of measurement

noise and errors. As an example, consider a modification to the preceding

equations, so that

3x1+2x -- 1,

x1- x2 -- Z,

ana 5x - $x = 8.

This may be written in matrix form as

3 2 x 1

I 2

- x2

-5 8


Page 8 - 16



Introduction to Seismic Inversion He.thods Brian Russell

'Now the third equation s not reducible to either of the other two, ana

an alternate solution must be found. The most popular aproach is the method

of least squares, which minimizes the sumof the squared error between the

solution and the observed results. That is, if we set the error to

e=Ax-y,

then we si reply mini mize

eTe--e , ez .......

n

, en = e ß

2

Le.

Re expressing the 'preceding equation in terms of the values x, y, and A,

we have

ß E = eTe (y - Ax)T(y Ax)

= yTy xTATyyTAx xTATAx.

We then solve the equation

bE_

bx

The final solution to the least-squares problem is given by the normal

equa i OhS

AT x = A y

or x = (ATA)-lATy

seismic data analysis

Documents