Stochastic vs Deterministic Pre- stack Inversion MethodsStochastic vs Deterministic Pre- stack...

Stochastic vs Deterministic Pre-stack Inversion Methods Brian Russell

Seismic reservoir analysis techniques utilize the fact that seismic amplitudes contain information about the geological properties of the reservoir.

The mathematics behind this observation was developed in the early 1900s, but its application to exploration seismic data did not start until the 1970s.

We classify these methods into two categories: methods that analyze only the amplitudes, and methods that invert the amplitudes to reservoir properties.

Newer methods analyze pre-stack data, where the analysis of the amplitudes without inversion is called Amplitude versus Offset, or AVO.

Pre-stack inversion has many forms, where the major division is between deterministic and stochastic, or geostatistical, methods.

In this talk I will discuss these methods and look at their assumptions and limitations.

Introduction

Seismic Data Well Log Data

Post-stack only

Gathers, only offsets

Gathers with azimuths

Post-stack inversion

AVO & Pre-stack

inversion

AVAz / Fracture Identification

Modeling for VS

VP, ρ VP, VS, ρ

Build rock physics model

Integrate using multivariate or Bayesian statistics

A suggested workflow

Inversion methods

Seismic Inversion Methods

Post-stack

Model Based

Recursive Sparse spike

Colored

Pre-stack Elastic

Impedance LMR

Simultaneous

Joint PP/PS Inversion

4D Inversion

Azimuthal Inversion

Stochastic / Geostatistical

Inversion

The basic model for inversion

The zero offset, or stacked, seismic trace can be modeled as the convolution of the acoustic impedance (AI) reflectivity with the wavelet.

As shown in the next slide, this is the basis for post-stack inversion.

Acoustic Impedance

Reflectivity

Wavelet W

Seismic

⇒= PVAI ρAIRWSW *=⇒⇒

∆AIAI = RAI 2

Post-stack inversion

6

Post-stack seismic Inversion, developed in the 1970s, reverses the forward modeling procedure, allowing us to derive the impedance from the reflectivity:

Impedance Reflectivity

Inverse Wavelet

Seismic

Qualitative AVO

In the 1980s, geophysicists observed that the amplitudes in a seismic gather could be written in linearized form using the amplitude versus offset (AVO) equation, a reformulation of the Aki-Richards linearized solution to the Zoeppritz equations:

Note that this has added two extra terms to the zero-offset case, a gradient term G and a curvature term C, often referred to as A, B and C, where the term A is called the intercept.

This formed the basis to what I refer to as qualitative AVO.

:where,sintansin)( 222 θθθθ CGRR AIP ++=

.2

and ,242

,22

22

P

P

P

S

S

S

P

S

P

P

P

PAI V

VCVV

VV

VV

VVG

VVR ∆

=∆

−

∆

−

∆=

∆+

∆=

ρρ

ρρ

Intercept and gradient analysis

Offset or Angle θ

The AVO equation predicts a linear relationship between these amplitudes and sin2θ. Regression curves are calculated to give RAI and G values for each time sample.

The amplitudes are extracted at all times, two of which are shown here:

Time

+RAI +G

- G

sin2θ

-RAI

Using the angle gathers for inversion

Fatti et al. (1994) re-formulated this equation to show that the pre-stack seismic data is a function of the acoustic impedance reflectivity (RAI), shear impedance reflectivity (RSI) and density reflectivity (RD) term:

,)( DSIAIP cRbRaRR ++=θ

,2

,,222

,2

whereρρρ

ρρ ∆

==∆

=∆

+∆

=∆

= DSS

SSIAI RVSI

SISI

VVR

AIAIR

.tansin4 and ,sin8,tan1 222

22

2 θθθθ −

=

−=+=

P

S

P

S

VVc

VVba

Independent pre-stack inversion Angle

Time (ms)

600

650 t

1 N Independent pre-stack inversion is

implemented by first extracting the reflectivity components, and then inverting them separately.

To estimate the reflectivities, the amplitudes at each time t in an N-trace angle gather are picked as shown here, to give RP(θ1)… RP(θN):

We can then solve for the reflectivities at each time sample using least-squares inversion.

Finally, these estimates are inverted using a post-stack type scheme.

=

−

)(

)( 11

NP

P

D

SI

AI

R

R

matrixweight

RRR

θ

θ

Reflectivities

Generalized inverse

Observations

Pre-stack inversion is also based on an extension of the Fatti formulation of the Aki-Richards equation:

operation. derivative theis and , angleat wavelet extracted the), and ,, of logarithms,, , angleat traceseismic)

:where,)()()()( 321

DW(θZZLLLS(θ

DLWcDLWcDLWcS

PPDSP

DSP

θρθ

θθθθ

===

++=

As in our discussion of AVO and independent inversion, this can again be set up as a least-squares problem:

model parameters = generalized inverse x observations As we discussed earlier, there are two main types of pre-stack inversion,

deterministic and stochastic.

Simultaneous Pre-stack Seismic inversion

Deterministic vs Stochastic Inversion

First of all, let us define the fundamental difference between deterministic and stochastic inversion: In deterministic inversion we produce what we consider to be a single

“best” solution. In stochastic inversion we produce many possible solutions, all

plausible, which average to the deterministic solution. The advantage of deterministic inversion is that we get the best “least-

squares” solution to our problem. The advantages of stochastic inversion are its higher frequency nature and

the calculation of uncertainty.

Deterministic pre-stack inversion example

On the next slide, I will show an example of deterministic pre-stack inversion.

– A Gulf Coast dataset (shown on the left of the slide) was inverted for P-impedance, S-impedance and density (which are shown on the right).

– The inverted volumes were transformed to Vshale, porosity and Sw (also shown on the right of the slide).

– Our assumption is that each inverted or transformed result is the “correct” answer.

– However, this will not allow us to obtain uncertainty estimates from of the rock properties.


Seismic Amplitude

Map

Inverted Acoustic

Impedance


Seismic Amplitude

Map

Inverted Acoustic

Impedance

Inverted Shear

Impedance


Seismic Amplitude

Map

Inverted Acoustic

Impedance

Inverted Shear

Impedance

Inverted Density


Seismic Amplitude

Map

Inverted Acoustic

Impedance

Inverted Shear

Impedance

Inverted Density

Derived Vshale Map

Derived Vshale


Seismic Amplitude

Map

Inverted Acoustic

Impedance

Inverted Shear

Impedance

Inverted Density

Derived Vshale Map

Derived Vshale

Derived Porosity


Seismic Amplitude

Map

Inverted Acoustic

Impedance

Inverted Shear

Impedance

Inverted Density

Derived Vshale Map

Derived Vshale

Derived Porosity

Derived Sw

Stochastic inversion In stochastic inversion, the least-squares inversion method is extended by

formulating the problem using a Gaussian or Log Gaussian posterior probability density function, or pdf (Tarantola, 1987).

This allows us to sample various scenarios from the pdf using the Monte Carlo (MC) or Markov Chain Monte Carlo (MCMC) approach.

The earliest approach to stochastic inversion was by Haas and Dubrule, 1994, in which Sequential Gaussian Simulation (SGS) is used.

Buland and Omre (2003) developed a fast approach to stochastic linearized inversion which utilized a Gaussian pdf.

The GeoSI method that I will discuss today combines both a Gaussian pdf and the SGS approach (Doyen, Williamson et al., 2007)

My colleague Ali Tehrani discussed the Jason StatMod approach yesterday.

Geostatistical inversion (Haas and Dubrulle)

AI simulations

Populate model with AI data at wells Define a random path through all (x,y) trace

locations At each trace location perform a local

optimization Generate a large number of trial AI

sequences using SGS with spatial and vertical variograms.

Compute reflectivity series and convolve with extracted wavelet.

Compute misfit against observed seismic.

Retain best matching AI (ρ >0.8). Go to next trace Adapted from

Dubrule, 2003

Actual seismic trace wavelet

Best simulated synthetic trace

(x,y)

Variogram models

Here are the variograms computed by Haas and Dubrulle (1994), showing the vertical, or temporal change, and the horizontal change including anisotropy.

Vertical (temporal) variogram

Horizontal variogram map showing anisotropy

Anisotropic variograms in principal directions

Bayesian stochastic inversion

Although geostatistical stochastic inversion produces reasonable results, it has two limitations: It is quite slow. It has difficulty in converging to an answer.

Buland and Omre (2003) introduced a new type of stochastic inversion which was based on multivariate Bayesian statistics. To illustrate the concept of Bayesian statistics, I will first consider the

standard least-squares regression problem. We will then look at the general theory proposed by Buland and Omre. We will then extend this method by combining it with SGS.

Least-squares regression

Consider a regression fit to 16 measured porosity values (φi) plotted against seismic impedance (zi), shown by the red line in the plot. This can be written:

ii bza +=φ

The regression line is the least-squares fit between porosity and impedance and is considered the “right” answer, even though only one point falls on it.

Statistical interpretation

The joint pdf p(φ,z) is the probability of φ and z occurring, and is defined by the variances and means, as well as the covariance between φ and z.

In the statistical interpretation of this crossplot, each variable (porosity and impedance) is given as a Gaussian probability distribution function (pdf) defined by its mean (µ) and variance (σ).

Joint pdf p(φ,z)

µz

σφ

µφ

σz

p(φ)

p(z)

Bayesian regression

The conditional mean µφ|z is the least-squares fit, and the conditional variance σφ | z gives us the “scatter” in this fit. Note it is narrower than p(φ).

)(),()|(

zpzpzp φφ =

conditional pdf p(φ | z)

σφ

σφ | z

µφ | z Bayesian statistics tells us that the conditional probability of φ given z, or the posterior, equals the joint probability divided by the probability of z, or the prior.

Bayesian stochastic inversion Generalizing the previous example to inversion, Buland and Omre (2003)

showed that:

)( 11|| mmd

Tdmdm CdCGC µµ −− +=

This equation reduces to the least-squares solution if we assume that µm = 0, and Cd = σd

2I:

dGCGGm Tmd

Tdm

112| )(ˆ −−+== σµ

.covariance lconditiona)( :and

(prior), mean model ,covariance model ,covariancedata mean, lconditiona

111|

|

=+=

==

==

−−−md

Tdm

mm

ddm

CGCGCC

Cµ

µwhere:

GeoSI

The GeoSI method, as implemented by CGG and ported to the Hampson-Russell suite of software, involves the following steps: Build a stratigraphic grid using horizons, well logs and layer-based kriging. Bring in partial angle stacks and wavelets. Compute the Bayesian posterior distribution by combining the model,

seismic data and well logs. Create multiple P and S-impedance realizations using the SGS technique. Compute the mean and standard deviations from the impedance

realizations. These steps are shown diagrammatically on the next two slides.

Building the stratigraphic grid

Horizons in time

Stratigraphic grid

Well logs in time (Vp, Vs, Density)

Low-pass filtering

Low-frequency prior model in stratigraphic grid

Ip

Is

Stratigraphic layering style

Layer-based Kriging

R. Moyen and J. Frelet

Stochastic Inversion Workflow

n Ip-Is realisations

Well logs (Vp, Vs, Density)

Well uncertainty

Partial angle stack seismic cubes

Ip-Is prior mean & standard deviation in stratigraphic grid

AI

time

Horizontal & vertical

variograms

Bayesian stochastic inversion

Posterior mean & standard deviation Ip-Is R. Moyen and J. Frelet

Bandwidth components For all inversion methods, the prior model is constructed by

interpolating filtered logs, and controls low frequencies. For both deterministic and stochastic inversion, the seismic

amplitudes control intermediate frequencies within the seismic bandwidth.

In stochastic inversion, the vertical variogram model controls the high frequencies.

Seismic Variogram model

Prior model

Frequency (Hz)

Pow

er

Spe

ctru

m

Adapted from Moyen and Frelet

Offshore West Africa example

Elastic inversion (Ip-Is) 3 seismic angle stacks

– 16°-30°-40° 120,000 traces Time window of 200 ms 132 layers in grid 500 realisations (59 Gb total) 3 wells with Vp, Vs and density logs Computations on standard workstation

Courtesy of R. Moyen and J. Frelet

Ip-Is Prior Model

2200 4200 m/s x g/cm3

5200 7200 m/s x g/cm3

P Impedance

S Impedance

200 m

s


Ip-Is Realisations

2200 4200 m/s x g/cm3

5200 7200 m/s x g/cm3

1 2 3 4

S Impedance

P Impedance

200 m

s


Ip Posterior Mean and Standard Deviation

5200 7200 m/s x g/cm3

Ip (km/s . g/cm3) σ 3.5 5.5

P Impedance std. dev.

P Impedance mean 200

ms


Vp/Vs Mean vs Realisations

Posterior mean 1.5 2.5 2.0

Realisation 1.5 2.5 2.0

1.5 2.5 Vp/Vs

Mean

Realisation

Sand/shale Cutoff

Inversion Results – Vp/Vs Ratio

Mean of 500 realisations

1.5 2.5

Vp/Vs

2.0


Using the realizations

One of the key questions about stochastic inversion is: what do we do with all the realizations?

In other words, wouldn’t a single answer (i.e. the deterministic solution) be better?

The answer is that with multiple realizations we can generate a number of new results, such as: Seismic lithology prediction. Facies classification. Volumetric uncertainty analysis. Petrophysical property analysis.

These concepts are illustrated in the next few slides.

Stochastic Lithology Prediction

N realisations of Ip, Is Histogram of sand volume

N sand / shale simulations

Ip

Is

Sand probability cube


P Impedance

Pois

son’

s ra

tio

3000 11000 0

0.5

VSH 1 0

Facies Discrimination

1.5 2.5

Ip/Is

2.0


Individual Realisations


Histogram of sand volume

For each realization, we can compute the sand volume from the number of cells with sand.

This can then be arranged in histogram format, and the probability percentiles can be computed. A percentile is computed from the total area under the probability

curve. Note that the percentile maps do not indicate a higher probability of

sand, only where the map falls within the distribution. As shown by an earlier slide, the percentile values will in general be

larger than mean computation. These concepts are illustrated in the next few slides.

Histogram of Sand Volume from Realizations

P10

P50

P90

Sand volume

Number of realisations

0

40

P10 P50 P90

Ranked lithology simulations

Sand Volume from Realisations and Mean

P10

P50

P90 Sand volume from inversion mean

Sand volume

Number of realisations

0

40

Connected Sand Geo-bodies

Color-code: geobody volume

Geobodies connected to at least one well (only largest are displayed)


Facies Probability from Stochastic Inversion

0.3 1 Sand probability

Large volume but small probability Smaller volume but

high probability Courtesy of R. Moyen and J. Frelet

Stochastic Petrophysical Modelling

Multiple Ip & Is models Multiple Vsh and Φ models

Ip

Φ

Statistical petro-elastic calibration


Geostatistical reservoir modeling – Interpolate between the wells

• Plausible details • Accurate near wells • Not elsewhere

Deterministic seismic inversion – Optimize P-Impedance to minimize synthetic-to-seismic misfit

• Accurate within seismic bandwidth • Unrealistically smooth • Only one possibility

StatMod/GeoSI geostatistical seismic inversion – Subsumes geostatistical modeling and deterministic inversion – Does both, simultaneously and in a statistically rigorous way

– Multiple plausible realizations at high detail (e.g. 1ms × 25m) – Yet also coherent “interpretations” of the seismic up to the km scale

Geostatistical inversion vs other modelling techniques

Conclusions

Stochastic inversion is a natural extension of deterministic inversion (mean of realizations ≈ deterministic inversion)

But it can provide extra information, such as:

Lithology probability

Facies distribution

Volumetrics

Petrophysical parameters

Our case study focussed on a channel sand play from West Africa.

Thank You

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