11-Geostatistical Methods for
11-Geostatistical Methods for
Seismic Inversion
Seismic Inversion
Amílcar Soares
Amílcar Soares
CERENA-IST
CERENA-IST
[email protected]
[email protected]
Seismic Seismic
Data Data
Seismic and Log Scale
Seismic and Log Scale
Acoustic Impedance Acoustic Impedance
Incident wave
Transmitted wave
Reflected wave Layer 1 impedance= Velocity(1) x Density(1) = Z1
Layer 2 impedance
= Velocity(2) x Density(2) = 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
Layered earth
Reflection Coefficients Impedance
Marine air gun Land dynamite
Time
Time (Sec.)
Time origin Zero phase
Lithology Impedance Minimum phase Zero phase Low velocity density
Lithology Impedance Zero phase wavelets High velocity density High velocity density Low velocity density
Incident wave
Reflected wave Layer 1 impedance= Velocity(1) x Density(1) = Z1
Layer 2 impedance
Impedance = Velocity X Density
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)
Convolving the reflectivity coefficients c(x) with a given wavelet w, one obtain the synthetic seismic
Earth Reflection
Coefficients Wavelet
Earth Reflection
Coefficients Wavelet
Wavelet Superposition Impedance
Earth Reflection
Coefficients Wavelet
Wavelet Superposition Impedance
Earth Reflection
Coefficients Wavelet
Wavelet Superposition Impedance
Earth Reflection Coefficients Wavelet Wavelet Superposition Recorded Trace Seismic Section Impedance
Recorded Trace Seismic
Reflection Coefficients Wavelet Recorded Trace Seismic Section
Reflection Coefficients Wavelet Recorded Trace Seismic Section Reflection Coefficients
-1000.0000 -500.0000 0.0000 500.0000 1000.0000 1500.0000 -20 -15 -10 -5 0 5 10 15 20 ms a m p l i t u d 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.
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 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
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 .
it is an iterative process based on the sequential simulation of trace values of acoustic impedances. -1000.0000 -500.0000 0.0000 500.0000 1000.0000 1500.0000 -20 -15 -10 -5 0 5 10 15 20 ms a m p l i t u d e
*
=
1- Choose randomly a trace to be generated. Simulation of N realizations of AI of that trace N Sinthetic tracerealizations 3-Compare with the real
seismic, choose and retain the best
realization Optimization algorithm 2- Convolution with a known wavelet
Geostatistical Inversion With Global
Perturbation Method
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
2- Convolution of transformed Simulated Acoustic Impedance -1000.0000 -500.0000 0.0000 500.0000 1000.0000 1500.0000 -20 -15 -10 -5 0 5 10 15 20 ms a m p l i t u d e
*
Impedancewith the real seismic a(x) obtaining local correlation coefficients cc(x)
5 – Return to step one to obtain a new
4 – From the N realizations, retain the traces with best matches and “compose” a best
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
transformation of images.
Let us consider that one wish to obtain a transformed image Z t (x), based on a set of Ni images Z 1(x), Z 2(x),…Z Ni (x),
with the same spatial dispersion statistics, e.g. variogram and global histogram: C (h) , (h) , F (z)
Direct co-simulation of Z t (x), having Z 1(x), Z 2(x),…Z Ni (x) as auxiliary variables, can be applied (Soares, 2001).
The collocated cokriging estimator of Z t (x) becomes:
( ) ( )
( ) ( )
) ( * ) ( 0 0 1 0 0 0 0 m x x Z x m x x Z x m x x Z i i Ni i i t t t t
The crossed correlograms
12(h) are calibrated by the
correlation coefficient between variables Z
1(x) and Z
2(x).
12 *(0):
)
(
.
)
0
(
)
(
12 * 1 12h
h
)
(
)
0
(
)
(
* 1 2h
h
=.95
=.80
=.60
following approximation is, in this case, quite appropriated:
The affinity of the transformed image Z t (x) with the multiple
images Z i (x) are determined by the correlation coefficients t,i (0). Remarks:
0 0 , , t t i t i t h h the corregionalization models are totally defined with the correlation coefficients t,i (0) between Z t (x) and Z i (x).
Assumption: to estimate Z t (x 0 ) the collocated value Z i (x 0 ) of a specific image Z i (x), with the highest correlation coefficient t,i (0), screens out the influence of the effect
of remaining collocated values Z j (x0), j i .
Hence, colocated co-kriging can be written with just one auxiliary variable : the “best” at location x 0:
( ) ( )
( ) ( )
) ( * ) ( x0 m x0 x0 Z x m x x0 Z x0 m x0 Z t t
t t i i i The “best” colocated data at x0.
1 i
...
(
)
(
)
(
)
(
)
)
(
*
)
(
x0 m x0 x0 Z x m x x0 Z x0 m x0 Z i i i t t t t
GSI – Global Stochastic Inversion
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.
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.
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 NCC 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
Direct S equential S imulation
1 – DSS 2 – CR & SY 3 – CC 4 – BCM & BAI 5 – DSco-SAI from wells
Variograms from wells
… N …
1 – DSS 2 – CR & SY 3 – CC 4 – BCM & BAI 5 – DSco-S … N … ) ( ) 1 ( ) ( ) 1 ( ) ( t Ai t Ai t Ai t Ai t Cr AI -40000 -20000 0 20000 40000 60000 80000 100000 120000 - 135 - 11 7 - 99 - 81 - 63 - 45 - 27 - 9 9 2 7 45 6 3 8 1 99 11 7 1 35
Wavelet
… N … SYSynthetic cubes
… N … CRCoefficient of
Reflection cubes
) ( ) ( ) (t Cr t wave z Sy Convolution1 – DSS 2 – CR & SY 3 – CC 5 – DSco-S … N … SY y x y x Y X Cov ( , ) ,
Real
seismic
cube
CC cube 4 – BCM & BAI… … N … … N 4 – BCM & BAI 1 – DSS 2 – CR & SY 3 – CC 5 – DSco-S … N … CC … N … AI
& & & & & &
Direct S equential co-Simulation
1 – DSS 2 – CR & SY 3 – CC 4 – BCM & BAI 5 – DSco-SAI from wells
Variograms from wells
… N …
AI
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 NCC 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
Seismic Data Set
Data extracted from a reservoir
Results from iteration 0 - Unconditional
Results from iteration 0 - Unconditional
Results from iteration 0 - Unconditional
Results from Process 0.85 0.87 0.88 0.80 0.62 0.08 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 C o r r e l a t i o n
Results from iteration 5
Results from iteration 5 Results from iteration 5
Results from iteration 5 Results from iteration 5
CC
Results from iteration 5 Results from iteration 5