11-Geostatistical Methods for Seismic Inversion · Geostatistical Seismic (Trace-by-Trace) Inversion (Bertolli et al, 1993): it is an iterative process based on the sequential simulation

11-Geostatistical Methods for

Seismic Inversion

Amílcar Soares

CERENA-IST

[email protected]

Seismic

Data

Seismic and Log Scale

01 - Introduction

Velocity X Density = AI

Acoustic Impedance

Recap: basic concepts

Incident wave

Transmitted wave

Reflected wave Layer 1 impedance

= Velocity(1) x Density(1)

= Z1

Layer 2 impedance


= Z2

Acoustic Impedance = Velocity X Density


“Since reflections are caused by changes in velocity and density, these two parameters are

combined into a parameter called “impedance”. This is the product of velocity and density “

Incident wave

Transmitted wave

Reflected wave

R = Reflected wavelet amplitude

Incident wavelet amplitude

R = Z2 - Z1

Z2 + Z1

R = (V2 x D2) - (V1 x D1)

(V2 x D2) + (V1 x D1)

Reflection coefficient


“ The ratio of the incident amplitude to the reflected amplitude is called the “Reflection Coefficient” .

Reflection coefficient can be seen a measure of the impedance contrast at the interface.”

Layered

earth

Reflection

Coefficients Impedance

Reflection coefficient Recap: basic concepts

Marine air gun Land dynamite

Time

C - 2

Wavelet Recap: basic concepts

“A wavelet is a wave-like oscillation with an amplitude that starts out at

zero, increases, and then decreases back to zero.”

Time (Sec.)

Time origin

Zero phase

Minimum phase


Lithology Impedance Minimum

phase

Zero

phase

Low

velocity

density

High

velocity

density

Recap: basic concepts Wavelet

Lithology Impedance

Zero phase

wavelets

High

velocity

density

High

velocity

density

Low

velocity

density


Incident wave

Transmitted wave

Reflected wave Layer 1 impedance


= Z1

Layer 2 impedance


= Z2

Impedance = Velocity X Density

02 – Seismic Inversion

Convolution

Incident wave

Transmitted wave

Reflected wave

Reflection coefficient

R = Reflected wavelet amplitude

Incident wavelet amplitude

R = Z2 - Z1

Z2 + Z1

R = (V2 x D2) - (V1 x D1)

(V2 x D2) + (V1 x D1)

02 – Seismic Inversion

Convolution

Layered

earth

Reflection


Convolution

Convolving the reflectivity coefficients c(x) with a

given wavelet w, one obtain the synthetic seismic

amplitudes a*(x)= c(x)*w

Principle of Seismic Inversion

Earth

Convolution - Forward exercise

Earth Impedance


Earth Reflection



Earth Reflection

Coefficients Wavelet Impedance


Earth Reflection

Coefficients Wavelet Wavelet

Superposition Impedance


Earth Reflection




Earth Reflection




Earth Reflection




Earth Reflection




Earth Reflection




Earth Reflection


Superposition

Recorded

Trace Impedance


Earth Reflection


Superposition

Recorded

Trace

Seismic

Section Impedance


Seismic

Section

Convolution - Inverse Exercise

Recorded

Trace Seismic

Section


Wavelet Recorded

Trace Seismic

Section


Reflection

Coefficients Wavelet Recorded

Trace Seismic

Section


Reflection


Trace Seismic

Section

Reflection

Coefficients


Reflection


Trace Seismic

Section

Reflection

Coefficients


Low Frequency Model

• Inverse Modeling is based on the physical relation:

-1000.0000

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0.0000

500.0000

1000.0000

1500.0000

-20 -15 -10 -5 0 5 10 15 20

ms

am

pli

tud

e

* =

Convolving the reflectivity coefficients c(x) with a

given wavelet w, one obtain the synthetic seismic

amplitudes a*(x)= c(x)*w

Typical Inverse Problem: one whish to know the acoustic impedances which

give rise to the known real seismic.

Typical Inverse Problem: one wish calculate the parameters ( high

resolution grid of acoustic impedance) that give rise to the solution we

know (the real seismic)

In this problem there is not a unique solution. One whish to find the set of

solutions that accomplish the spatial requisites of the acoustic impedance

grid: spatial continuity pattern, global CDfs, ...

Outline of the iterative method

Space of the

Parameters

Solution for

the set of

parameters

Compare with the

known real solution

Is the match

satisfactory ?

N

Change the set of

parameters in order to

make the process

convergent

Geostatistical Seismic Inversion

The aim of geostatistical inversion of seismic is to produce high

resolution of numerical models that have two properties:

•The numerical model honors a physical relationship (convolution model)

with the actual data .

•The numerical model reflects the spatial continuity and the global

distribution functions .

Geostatistical Seismic (Trace-by-Trace) Inversion (Bertolli et al, 1993):

it is an iterative process based on the sequential simulation of trace values of acoustic

impedances.

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ms

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* =

1- Choose randomly

a trace to be

generated.

Simulation of N

realizations of AI of

that trace

N Sinthetic

trace

realizations 3-Compare with the real

seismic, choose and

retain the best

realization

4- return until all traces

are simulated

Optimization

algorithm

2- Convolution

with a known

wavelet

GSI – Global Stochastic Inversion

Geostatistical Inversion With Global

Perturbation Method

Part I - Theory

The approach of Global Stochastic

Inversion is based on two key ideas:

•the use of the sequential direct co-

simulation as the method of

“transforming” 3D images, in a iterative

process and

•to follow the sequential procedure of

the genetic algorithms optimization to

converge the transformed images

towards an objective function


2- Convolution of transformed Simulated

Acoustic Impedance

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0.0000

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-20 -15 -10 -5 0 5 10 15 20

ms

am

pli

tud

e

*

1 – Simulation of Acoustic

Impedance

3 – Comparing the synthetic amplitudes a*(x)

with the real seismic a(x) obtaining local

correlation coefficients cc(x)

5 – Return to step one to obtain a new

generation of AI images until a given objective

function is reached.

4 – From the N realizations, retain the traces

with best matches and “compose” a best

image of AI

An iterative inversion methodology is proposed based on a

direct sequential simulation and co-simulation approaches:

•Several realizations of the entire 3D cube of acoustic impedances are simulated in a first step, instead individual traces or cells;

•After the convolution local areas of best fit of the different images are selected and “merged” into a secondary image of a direct co-simulation in the next iteration;

•The iterative and convergent process continues until a given match with objective function is reached. Spatial dispersion and patterns of acoustic impedances (as revealed by histograms and variograms) are reproduced at the final acoustic impedance cube.

•In a last step, porosity images are derived from the seismic impedances and the uncertainty derived from the seismic quality is assessed based on the quality of match between synthetic seismogram obtained by seismic inversion and real seismic.

The use of Direct Sequential Co-Simulation for global transformation of images.

Let us consider that one wish to obtain a transformed image Zt(x), based on a set of Ni images Z1(x), Z2(x),…ZNi(x),

with the same spatial dispersion statistics, e.g. variogram and global histogram: C (h) , (h) , F (z)

Direct co-simulation of Zt(x), having Z1(x), Z2(x),…ZNi(x) as auxiliary variables, can be applied (Soares, 2001).

The collocated cokriging estimator of Zt(x) becomes:

)()()()()(*)( 00

1

0000 xmxZxxmxZxxmxZ ii

Ni

i

itttt

Colocated data of Ni

secondary images

Variable Z1(x)

3 realizations from variable Z2(x)

“Markov-type” approximation:

The crossed correlograms 12(h) are calibrated by the

correlation coefficient between variables Z1(x) and Z2(x).

12*(0):

)(.)0()( 1

*

1212 hh

)()0(

)0()( 12*

12

*

1212 hh

global

global

=.95 =.80 =.60

Variable Z1(x)

Simulation of variable Z2(x)

Since the models i(h), i=1, Ni, and t(h) are the same, the following approximation is, in this case, quite appropriated:

The affinity of the transformed image Zt(x) with the multiple

images Zi(x) are determined by the correlation coefficients t,i(0).

Hence, one can select the images which characteristics we wish

to “preserve” in the transformed image Zt(x)

Remarks:

0

0,,

t

titit

hh

the corregionalization models are totally defined with the correlation coefficients t,i(0) between Zt(x) and Zi(x).

Local Screening Effect Approximation

Assumption: to estimate Zt(x0) the collocated value Zi(x0) of a specific image Zi(x),

with the highest correlation coefficient t,i(0), screens out the influence of the effect

of remaining collocated values Zj(x0), j i.

Hence, colocated co-kriging can be written with just one auxiliary variable : the “best” at location x0:

)()()()()(*)( 000000 xmxZxxmxZxxmxZ iiitttt

The “best” colocated

data at x0.

)()()()()(*)( 00

1

0000 xmxZxxmxZxxmxZ ii

Ni

i

itttt

...

)()()()()(*)( 000000 xmxZxxmxZxxmxZ iiitttt

The “best” colocated data at x0:

highest Correlation Coeffificient

t,i(0) .

Outline of the proposed methodology


i- Generate a set of initial images of acoustic impedances by using direct sequential simulation.

ii- Create the synthetic seismogram of amplitudes, by

convolving the reflectivity, derived from acoustic impedances, with a known wavelet.

iii- Evaluate the match of the synthetic seismograms, of entire

3D image, and the real seismic by computing, for example local correlation coefficients.

iv - Ranking the “best” images based on the match (e.g. the average value or a percentile of correlation coefficients for the entire image). From them, one select the best parts- the columns or the horizons with the best correlation coefficient – of each image. Compose one auxiliary image with the selected “best” parts, for the next simulation step. v- Generate a new set of images, by direct co-simulation, and return to step ii) until a given threshold of the objective function is reached.

AI from wells

N stochastic simulations

of AI based upon well data and variograms.

Calculation of Coefficients of Reflection (CR)

Calculation of the N Synthetic cubes:

convolution of CR cubes with a wavelet.

Calculation of Correlation Coefficient (CC)

between the synthetics and the seismic cubes.

A new CC map (Best Correlation Map, BCM) and the

corresponding AI secondary image (Best AI, BAI) are

created:

The highest CC of the N CC maps is allocated to each

x0 location.

The corresponding AI values are used to build the BAI

cube to be used as secondary data set.

N stochastic co-simulations (DSco-S) of AI based

upon well data and conditioned to BCM.

3D seismic

cube

n iterations

Wavelet

03 – Algorithm Description

Algorithm Description

Direct Sequential Simulation

1 – DSS

2 – CR & SY

3 – CC

4 – BCM &

BAI

5 – DSco-S

AI from wells Variograms from wells

… N …

Simulated cubes of AI

AI


1 – DSS

2 – CR

& SY

3 – CC

4 – BCM

& BAI

5 – DSco-S

… N …

)()1(

)()1()(

tAitAi

tAitAitCr

AI

-40000

-20000

0

20000

40000

60000

80000

100000

120000

-135 -117 -99 -81 -63 -45 -27 -9 9 27 45 63 81 99 117 135

Wavelet

… N … SY

Synthetic cubes

… N … CR Coefficient of

Reflection cubes

)()()( zwavetCrtSy Convolution


1 – DSS

2 – CR

& SY

3 – CC

5 – DSco-S

… N … SY

yx

yx

YXCov

),(,

Real

seismic

cube

CC

cube

… N … CC Correlation cubes

4 – BCM

& BAI


… …

N

… …

N

4 – BCM

& BAI

1 – DSS

2 – CR

& SY

3 – CC

5 – DSco-S

… N … CC

… N … AI & & & & & &

BCM BAI


Direct Sequential co-Simulation

1 – DSS

2 – CR

& SY

3 – CC

4 – BCM

& BAI

5 – DSco-S

AI from wells Variograms from wells

… N …

Simulated cubes of AI

AI

BCM BAI


AI from wells

N stochastic simulations

of AI based upon well data and variograms.

Calculation of Coefficients of Reflection (CR)

Calculation of the N Synthetic cubes:

convolution of CR cubes with a wavelet.

Calculation of Correlation Coefficient (CC)

between the synthetics and the seismic cubes.

A new CC map (Best Correlation Map, BCM) and the

corresponding AI secondary image (Best AI, BAI) are

created:

The highest CC of the N CC maps is allocated to each

x0 location.

The corresponding AI values are used to build the BAI

cube to be used as secondary data set.

N stochastic co-simulations (DSco-S) of AI based

upon well data and conditioned to BCM.

3D seismic

cube

n iterations

Wavelet


04 – Results

Seismic Data Set

Data extracted from a

reservoir

Interpreted Horizons to

quality control

Variograms

Histogram, basic statistics and Wavelet

Wells

From 19 only 2 had Velocity log

04 – Results

Wells

04 – Results

Wells

04 – Results

Wells – Histogram and Basic Statistics

Acoustic Impedance

04 – Results

Results from iteration 0 - Unconditional

AI from Simulation 1 AI from Simulation 15

04 – Results


SY from Simulation 1 SY from Simulation 15

04 – Results


CC from Simulation 1 CC from Simulation 15

04 – Results


Average from Simulations Standard Deviation from

Simulations

04 – Results


Best Acoustic Impedance cube Best Correlation Cube

04 – Results

Results from Process

0.85 0.87 0.88

0.80

0.62

0.080

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 1 2 3 4 5

Iterations

Co

rre

lati

on

04 – Results

Results from iteration 5

AI from Simulation 3 AI from Simulation 28

04 – Results


SY from Simulation 3 SY from Simulation 28

04 – Results


CC from Simulation 3 CC from Simulation 28

04 – Results


Average from Simulations Standard Deviation from

Simulations

Good match with the horizons in the final AI cube

04 – Results

Synthetic Seismic Real Seismic

Practice VII- Seismic Inversion

Practice with GSI (Student) – Global Stochastic Inversion

Practice with S-GeMS – Stanford Geostatistical Modelling Software

Documents

11-Geostatistical Methods for Seismic Inversion · Geostatistical Seismic (Trace-by-Trace) Inversion (Bertolli et al, 1993): it is an iterative process based on the sequential simulation