11-Geostatistical Methods for Seismic Inversion Amílcar Soares CERENA-IST
[email protected]
01 - Introduction Introducti on
Seismic and Log Scale
Seismic Data
Recap: basic concepts
Acoustic Impedance
Velocity
X
Density =
AI
Recap: basic concepts Acoustic Impedance = Velocity X Density
Reflected wave Incident wave
Transmitted wave
Layer 1 impedance = Velocity(1) x Density(1) = Z1
Layer 2 impedance = Velocity(2) x Density(2) = Z2
“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 “
Recap: basic concepts
Reflection coefficient
Reflected wave Incident wave
R=
R=
R=
Reflected wavelet amplitude Incident wavelet amplitude Z2 - Z1 Z2 + Z1 (V2 x D2) - (V1 x D1) (V2 x D2) + (V1 x D1)
Transmitted wave
“ 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.”
Recap: basic concepts
Reflection coefficient Layered earth
Impedance
Reflection Coefficients
Recap: basic concepts
Wavelet Land dynamite
Marine air gun
Time “A wavelet is a wave-like oscillation with an amplitude that start s out at zero, increases, and then decreases back to zero.” C-2
Recap: basic concepts
Wavelet
Minimum phase
Time (Sec.)
Zero phase
Time origin
Recap: basic concepts
Lithology
Low velocity density
High velocity density
Wavelet Impedance
Minimum phase
Zero phase
Recap: basic concepts
Lithology
High velocity density
Low velocity density
High velocity density
Wavelet Impedance
Zero phase wavelets
02 – S eis mic Invers ion Convolution Impedance = Velocity X Density
Reflected wave Incident wave
Transmitted wave
Layer 1 impedance = Velocity(1) x Density(1) = Z1
Layer 2 impedance = Velocity(2) x Density(2) = Z2
02 – S eis mic Invers ion Convolution
Reflection coefficient Reflected wave Incident wave
R=
R=
R=
Transmitted wave
Reflected wavelet amplitude Incident wavelet amplitude Z2 - Z1 Z2 + Z1 (V2 x D2) - (V1 x D1) (V2 x D2) + (V1 x D1)
Convolution Layered earth
Impedance
Reflection Coefficients
Principle of Seismic Inversion
Convolving the reflectivity coefficients c(x) with a given wavelet w, one obtain the synthetic seismic amplitudes a*(x)= c(x) *w
Convolution - Forward exercise Earth
Convolution - Forward exercise Earth
Impedance
Convolution - Forward exercise Earth
Impedance
Reflection Coefficients
Convolution - Forward exercise Earth
Impedance
Reflection Coefficients
Wavelet
Convolution - Forward exercise Earth
Impedance
Reflection Coefficients
Wavelet
Wavelet Superposition
Convolution - Forward exercise Earth
Impedance
Reflection Coefficients
Wavelet
Wavelet Superposition
Convolution - Forward exercise Earth
Impedance
Reflection Coefficients
Wavelet
Wavelet Superposition
Convolution - Forward exercise Earth
Impedance
Reflection Coefficients
Wavelet
Wavelet Superposition
Convolution - Forward exercise Earth
Impedance
Reflection Coefficients
Wavelet
Wavelet Superposition
Convolution - Forward exercise Earth
Impedance
Reflection Coefficients
Wavelet
Wavelet Superposition
Convolution - Forward exercise Earth
Impedance
Reflection Coefficients
Wavelet
Wavelet Superposition
Recorded Trace
Convolution - Forward exercise Earth
Impedance
Reflection Coefficients
Wavelet
Wavelet Superposition
Recorded Trace
Seismic Section
Convolution - Inverse Exercise Seismic Section
Convolution - Inverse Exercise Seismic Section
Recorded Trace
Convolution - Inverse Exercise Seismic Section
Recorded Trace
Wavelet
Convolution - Inverse Exercise Seismic Section
Recorded Trace
Wavelet
Reflection Coefficients
Convolution - Inverse Exercise Seismic Section
Recorded Trace
Wavelet
Reflection Coefficients
Reflection Coefficients
Convolution - Inverse Exercise Seismic Section
Recorded Trace
Wavelet
Reflection Coefficients
Reflection Coefficients
Low Frequency Model
•
Inverse Modeling is based on the physical relation: Convolving the reflectivity coefficients c(x) with a given wavelet w, one obtain the synthetic seismic amplitudes a*(x)= c(x) *w
1500.0000
1000.0000
*
-20
=
500.0000
e d u t i l p m a
0.0000 -15
-10
-5
0
5
10
15
20
-500.0000
-1000.0000 ms
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) Outline of the iterative method
Space of the Parameters Change the set of parameters in order to make the process convergent
Solution for the set of parameters
Compare with the known real solution Is the match satisfactory ?
N
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, ...
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.
1500.0000
1000.0000
*
-20
0.0000 -15
-10
-5
0
5
10
15
-500.0000
-1000.0000
1- Choose randomly a trace to be generated. Simulation of N realizations of AI of that trace
Optimization algorithm
500.0000
e d u t i l p m a
ms
2- Convolution with a known wavelet
20
= N Sinthetic trace realizations
4- return until all traces are simulated
3-Compare with the real seismic, choose and retain the best realization
GSI – Global Stochastic Inversion Geostatistical Inversion With Global Perturbation Method
Part I - Theory
GSI – Global Stochastic Inversion
The approach of Global Stochastic Inversion is based on two key ideas: •the use of the sequential direct cosimulation 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
1 – Simulation of Acoustic Impedance 2- Convolution of transformed Simulated Acoustic Impedance
1500.0000
1000.0000
*
500.0000
e d u t i l p m a -20
0.0000 -15
-10
-5
0
-500.0000
-1000.0000 ms
5
10
15
20
3 – Comparing the synthetic amplitudes a*(x) with the real seismic a(x) obtaining local correlation coefficients cc(x)
4 – From the N realizations, retain the traces with best matches and “compose” a best image of AI
5 – Return to step one to obtain a new generation of AI images until a given objective function is reached.
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 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: Ni
Z t ( x 0 ) * mt ( x 0 ) x 0 Z t ( x ) mt ( x ) i x 0 Z i ( x 0 ) mi ( x 0 )
i 1
Colocated data of Ni secondary images
Variable Z1(x)
3 realizations from variable Z 2(x)
“Markov-type” approximation:
The crossed correlograms 12 (h) are calibrated by the correlation coefficient between variables Z 1(x) and Z2(x). 12 * (0):
12 (h) 12 (0)* . 1 (h)
12 ( h )
1 2 ( 0) 12
g lo b a l
*
( 0)
*
12
g lob a l
( h)
Simulation of variable Z 2(x)
Variable Z1(x)
=.95
=.80
=.60
Since the models i (h), i=1, Ni, and t(h) are the same, the following approximation is, in this case, quite appropriated: t ,i h t ,i 0
t h t 0
the corregionalization models are totally defined with the correlation coefficients t,i (0) between Z t (x) and Z i (x).
Remarks: 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).
Hence, one can select the images which characteristics we wish to “preserve” in the transformed image Z t (x)
Local Screening Effect Approximation
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: Z t ( x 0 ) * mt ( x 0 )
x0 Z ( x ) m ( x ) x0 Z ( x0 ) m ( x0 ) t
t
i
i
i
The “best” colocated data at x0.
Ni
Z t ( x0 ) * mt ( x0 ) x0 Z t ( x ) mt ( x ) i x0 Z i ( x 0 ) mi ( x0 ) i 1
...
Z t ( x0 ) * mt ( x0 )
x0 Z ( x ) m ( x ) x0 Z ( x0 ) m ( x0 ) t
t
i
i
i
The “best” colocated data at x0: highest Correlation Coeffificient t,i (0) .
Outline of the proposed methodology 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.
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.
03 – A lg orithm Des cription Algorithm Description N stochastic simulations of AI based upon well data and variograms.
n iterations
Calculation of Coefficients of Reflection (CR) Calculation of the N Synthetic cubes: convolution of CR cubes with a wavelet.
Wavelet
Calculation of Correlation Coefficient (CC) between the synthetics and the seismic cubes.
3D seismic cube
A new CC map (Best Correlation Map, BCM) and the corresponding AI secondary image (Best AI, BAI) are created:
AI from wells 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.
Algorithm Description
AI from wells
Variograms from wells
1 – DSS
2 – CR & SY
3 – CC
Direct S equential S imulation
4 – BCM & BAI
5 – DSco-S
AI
…N…
Simulated cubes of AI
Algorithm Description
1 – DSS
AI
…N…
2 – CR & SY
3 – CC
Cr (t )
CR
4 – BCM & BAI
Ai(t 1) Ai(t ) Ai(t 1) Ai(t )
…N…
Convolution
Coefficient of Reflection cubes
Sy(t ) Cr (t ) wave( z )
120000
100000
80000
60000
40000
20000
5 – DSco-S
0 - 1 35
- 11 7
- 9 9
- 8 1
- 3 6
- 4 5
- 2 7
- 9
9
2 7
45
6 3
8 1
99
-20000
SY
…N…
Synthetic cubes
-40000
Wavelet
11 7
1 3 5
Algorithm Description SY
…N…
1 – DSS
2 – CR & SY
3 – CC x, y
Cov( X , Y )
x y
4 – BCM & BAI
CC cube
5 – DSco-S
CC
…N…
Real seismic cube Correlation cubes
Algorithm Description AI
…N… &
&
CC
&
&
&
&
…N…
1 – DSS
2 – CR & SY
3 – CC
4 – BCM & BAI
5 – DSco-S
…
…
N…
N…
BCM
BAI
Algorithm Description
AI from wells 1 – DSS
Variograms from wells BCM
BAI
2 – CR & SY
3 – CC
Dir ect S equential co-Simulation
4 – BCM & BAI
5 – DSco-S
AI
…N…
Simulated cubes of AI
Algorithm Description N stochastic simulations of AI based upon well data and variograms.
n iterations
Calculation of Coefficients of Reflection (CR) Calculation of the N Synthetic cubes: convolution of CR cubes with a wavelet.
Wavelet
Calculation of Correlation Coefficient (CC) between the synthetics and the seismic cubes.
3D seismic cube
A new CC map (Best Correlation Map, BCM) and the corresponding AI secondary image (Best AI, BAI) are created:
AI from wells 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.
04 – R es ults 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 – R es ults Wells
04 – R es ults Wells
04 – R es ults Wells – Histogram and Basic Statistics
Acoustic Impedance
04 – R es ults Results from iteration 0 - Unconditional
AI from Simulation 1
AI from Simulation 15
04 – R es ults Results from iteration 0 - Unconditional
SY from Simulation 1
SY from Simulation 15
04 – R es ults Results from iteration 0 - Unconditional
CC from Simulation 1
CC from Simulation 15
04 – R es ults Results from iteration 0 - Unconditional
Average from Simulations
Standard Deviation from Simulations
04 – R es ults Results from iteration 0 - Unconditional
Best Acoustic Impedance cube
Best Correlation Cube
04 – R es ults Results from Process
1 0.9 0.85
0.8
0.87
0.88
0.80
0.7 n o 0.6 i t a l 0.5 e r r o 0.4 C
0.62
0.3 0.2 0.1 0.08 0 0
1
2
3 Iterations
4
5
04 – R es ults Results from iteration 5
AI from Simulation 3
AI from Simulation 28
04 – R es ul ultts Results from iteration 5
SY from Simulation 3
SY from Simulation 28
04 – R es ul ultts Results from iteration 5
CC from Simulation 3
CC from Simulation 28
04 – R es ul ultts Results from iteration 5
Average from Simulations
Standard Deviation from Simulations
Good match with the horizons in the final AI cube
04 – R es ults
Synthetic Seismic
Real Seismic
