2 Post Stack Inversion

2-1

Seismic Inversion and AVO applied to Lithologic Prediction

Part 3Post-stack inversion theory

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Introduction to Post-Stack Seismic Inversion

• Recall from our initial introduction that we will consider the following inversion methods in this discussion:

Post-stack Amplitude Inversion

Bandlimited Inversion

Sparse-Spike Inversion

Model-based Inversion

• An alternate name for bandlimited inversion is recursive inversion, which will be discussed first.

• Before discussing inversion, we will discuss the convolutional model.

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The Convolutional ModelThe seismic trace is modeled as follows:

s(t) = w(t)*r(t) + n(t)

or: seismic = wavelet * reflectivity + noisewhere * means convolution.

The assumptions are that the data is: Post-stack data - the traces are zero-offset There are no multiples There are no AVO effects The noise is random, white noise, uncorrelated with the

seismic. There is no coherent noise. The wavelet is constant - not varying with time. The seismic data are already migrated - each seismic trace

depends only on the reflectivity sequence directly below the seismic trace location.

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The Convolutional Model

The synthetic to the left illustrates the equation of the previous page for the noise-free case. Notice that convolution with the wavelet results in a loss of information.

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The Convolutional ModelIn the frequency domain, convolution is the product of the reflectivity spectrum and the wavelet spectrum, or S(f) = W(f)R(f)

This means that the seismic trace has lost both the high frequency and the low frequency portions of the spectrum.

Deconvolution and inversion attempt to recover these lost regions. The method of filling in the missing data depends on the inversion algorithm.

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The Earth’s Reflectivity

- This equation is true only for vertical incidence rays. This means that AVO effects are assumed to be negligible.- If the seismic data does contain an AVO anomaly, the inversion process will (erroneously) attribute the entire effect to changes in density and velocity, rather than to changes in Poisson’s Ratio.- Since the reflection coefficient depends only on the product of density and velocity, and not on either individually, it follows that post-stack inversion is incapable of solving separately for density or velocity. Only impedance changes can be measured by inversion.

wave.- Sor P either i, layer ofvelocity =V

i layer ofdensity = :where

i

i

Before discussing wavelet effects, let us look only at the reflectivity of the earth, which consists of all the reflection coefficients. The reflection coefficient for the ith interface is defined as:

ii1i1i

ii1i1ii VV

V - V = R

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Recursive Inversion

Recursive Inversion, also called bandlimited inversion, is the simplest form of inversion. The reflection coefficient for the ith interface of N layers can be written:

i1i

i1ii Z Z

Z- Z = R

i

ii1i R- 1

R + 1Z = Z

Therefore, the impedance of the i +1st layer can be determined from the ith layer:

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Recursive Inversion

Seismic raypath

Interface at depth = d

Z1 = 1V1

Z2 = 2V2

Rt

Reflection at time t = 2d/V1

GeologyGeology SeismicSeismic

For a single layer, as shown above, we can write:

R- 1R + 1Z = Z

Z ZZ- Z = R 12

12

12

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Example from first exercise

Recall that we calculated the following wet and gas cases in the first section using the Biot-Gassmann equations:

Wet: VP= 2500 m/s, VS= 1250 m/s, = 2.11 g/cc

Gas: VP= 2000 m/s, VS= 1310 m/s, = 1.95 g/cc

Let us add a shale with the following properties:

VP= 2250 m/s, VS= 1125 m/s, = 2.0 g/cc

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Exercise 2-1(A) Using the values on the previous slide, compute the P

and S-impedances for the three cases:

ZPgas= ZPwet=

ZSgas= ZSwet=

ZPshale= ZSshale=

(B) Compute the P and S reflection coefficients for shale over sand for the gas and wet cases:

RPgas= RSgas=

RPwet= RSwet=

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Exercise 2-1 Answers

(A) The P and S-impedances for the three cases are:

ZPgas= 3900 m/s*g/cc ZSgas= 2555

ZPwet= 5275 ZSwet= 2638

ZPshale= 4500 ZSshale= 2250

(B) The P and S reflection coefficients for shale over sand for the gas and wet cases are:

RPgas= -0.071 RSgas= 0.063

RPwet= 0.079 RSwet= 0.079

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Exercise 2-1(C) Using the formula:

compute the gas and wet impedances for the P and S-wave cases using the computed shale impedance for the first layer and the reflection coefficients computed in the first part of the exercise

ZPgas=

ZPwet=

ZSgas=

ZSwet=

R- 1R+ 1Z= Z 12

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Exercise 2-1 Answers(C) The gas and wet impedances for the P and S-wave

cases are:

39000.071 10.071- 14500

R- 1R+ 1

Z= ZPgas

PgasPshalePgas

52750.079 10.079 14500

R- 1R+ 1Z= Z

Pwet

PwetPshalePwet

25530.063 10.063 12250

R- 1R+ 1

Z= ZSgas

SgasSshaleSgas

26380.079 10.079 12250

R- 1R+ 1Z= Z

Swet

SwetSshaleSwet

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Recursive Inversion of N layers

terms. all of product the :where

,r1r1

N

1N

1i i

i1z =z

Π

Notice that for N layers, we can start at the first layer and compute the impedance of each successive layer by recursively applying this formula:

This is illustrated in the next slide.

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Recursive Inversion

Applying recursive inversion under ideal conditions, we can perfectly recover the impedance, as shown on the right:

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Recursive Inversion of N layers

In fact, there are two problems that are not illustrated in the ideal case shown in the previous slide:

- First, if there are errors in the reflectivity from noise poor scaling, errors in the inversion will be cumulative, and will get worse with time.

- Second, as will now be shown, a bigger problem is with the convolution of the seismic wavelet. To illustrate this problem, we will start with a simple, but instructive, exercise.

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Exercise 2-2

Using the recursive inversion formula given earlier, and assuming that = 1 (i.e. V = Z) and V1 = 1000 m/s, work out the inverted velocities for (a) a single reflector, and (b) the reflector convolved with a Ricker wavelet, as shown on the left.

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Note that we can only recover the true value for the change in impedance if we have a single spike, which is not the case after convolution with the wavelet.

V2 = 818 m/s

V2 = 1500 m/s

V1 = 1000 m/s

V3 = 1227 m/s

V4 = 1004 m/s

V1 = 1000 m/s

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The Problem with Recursive Inversion

First, the wavelet “lobes” cause low and high impedance zones to appear on the inverted trace which are not geologically valid.

Second, the low frequency component of the impedance is lost.

Third, the true impedance value is never estimated due to the first two problems.

As shown in the next exercise, the problem gets worse with multiple reflections.

As shown in the previous exercise, there are three problems with recursive inversion:

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Exercise 2-3: More than one layer

Consider the impedance model shown below, with interfaces at arbitrary times t1 and t2. Compute the resulting reflectivity:

1000 1250 1500750 1750

Acoustic Impedance (m/s*g/cc)

t1

t2

Z1 = 1223

Z2 = 1000

Z3 = 1500

Reflectivity

0 0.1 0.2-0.1-0.2

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Exercise 2-3: Answer


1000 1250 1500750 1750


t1

t2

Z1 = 1223

Z2 = 1000

Z3 = 1500

Reflectivity

0 0.1 0.2-0.1-0.2

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Exercise 2-4: Convolving with a wavelet

In the previous example the computed reflection coefficients are r = (-0.1, 0.2). Letting the wavelet be w = (-1, 2, -1), and the reflectors be three samples apart, we can convolve w with r to produce the seismic trace using the following matrix multiply. Graph the seismic trace to the right:

2.04.02.0

1.02.0

1.0

00

0.2000.100

121000000121000000121000000121000000121000000121

Seismic Trace0 0.2 0.4-0.2-0.4

t

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The result looks like this:


t First scaledwavelet

Second scaledwavelet

Exercise 2-4 Answer

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Notice that the trace looks like the one below, and both wavelets are separate. In other words, convolution has scaled the wavelet by the value of each reflection coefficient, and shifted the scaled wavelet to the location of the reflection coefficient.




Convolving with a wavelet

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The effect of zeros in the convolution

In the previous example we could have ignored the zeros at the beginning and end of the reflectivity and got the same result. Notice that this is equivalent to dropping the same column number in the matrix as the row of the zero, as shown below:

0.20.40.2

0.10.2

0.1

0.2000.1

100021001210

012100120001

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Exercise 2-5Now, we will drop the zeros between the reflection coefficients, which means the coefficients are one sample apart. Re-compute the seismic trace using the matrix method just described. Use the box on the left for your calculations. Draw the new trace on the right, putting zeros in the first and last samples. How does this new trace compare with the original trace? Seismic TraceSeismic Trace

0 0.2 0.4-0.2-0.4

t

____

0.20.1

102112

01

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Exercise 2.5 Answer

Note that the resulting trace is “tuned”, meaning that the two zero-phase wavelets have been merged into a single ninety-degree wavelet.

2.05.04.0

1.0

2.01.0

102112

01


t

0.5

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Convolving with a wavelet

Notice that the new trace looks like the one below, and both wavelets have “tuned” together. The result looks like a ninety degree phase wavelet rather than the original zero phase wavelet. (Phase will shortly be discussed).

Tunedwavelet


t

0.5

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Practical inversion schemes

Now that we have looked at the issues that make inversion difficult, such as noise problems, wavelet tuning, and the fact that recursive inversion only works if a wavelet is not present, let us look at practical schemes for inversion.

We will look at the following steps:

(1) Wavelet extraction(2) Geological model-building(3) Integrating the geological model with the inverted seismic data.

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Wavelets and wavelet extraction

As shown in the next figure, the wavelet is defined completely by its amplitude and phase spectra.

- Over a limited frequency range, the phase spectrum may often be approximated by a straight line.- The intercept of the line is the constant phase rotation which best characterizes this wavelet.-The slope of the line measures the time-shift of the wavelet.

The wavelet on the next page is an example of a minimum phase wavelet.

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A typical wavelet

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Wavelets in the earth vary both laterally(spatially) and temporally for a variety of reasons:

Near surface effects (space variant) Frequency-dependent absorption (space and time variant) Inter-bed multiples (space and time variant) NMO stretch Processing artifacts

We usually assume that the wavelet is constant with time and space: Time invariant : This means that the inversion is optimized

for a limited time window. Space invariant : This assumes that the data has been

processed optimally to remove spatial variations in the wavelet.

Wavelet extraction

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The following methods can be used for wavelet extraction:

(1) Estimate amplitude spectrum using the the seismic data alone. The phase is assumed known from some other source. Methods include autocorrelation, maximum entropy spectral analysis, and cross spectral analysis.

(2) Estimate both amplitude and phase spectra from the seismic data alone. Methods include minimum entropy wavelet estimation and higher order moments. Note that these methods can be quite unstable.

(3) Estimate both amplitude and phase spectra using deterministic measurements, such as marine signatures and VSP analysis.

(4) Estimate both amplitude and phase spectra using both seismic and well log measurements.

(5) Estimate amplitude spectrum and a constant phase spectrum using both seismic and well log measurements.

Wavelet extraction

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A zero-phase seismic wavelet

(a) The wavelet estimated using from the amplitude spectrum of the seismic data.

(b) The amplitude spectrum of the wavelet.

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A general problem with wavelet extraction is that:

• To extract a wavelet using logs, an optimum correlation must be done first.

• To perform correlation properly, the wavelet must already be known.

A practical wavelet extraction procedure is as follows:

(1) Use statistical wavelet extraction to determine a preliminary wavelet. This assumes that the approximate phase of the wavelet is known.

(2) Stretch/squeeze the logs to tie the seismic data

(3) Extract a new wavelet using the well logs.

(4) Possibly repeat steps (2) and (3).

Wavelet Extraction

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Non-uniqueness – why we need a model• All inversion algorithms suffer from the “non-uniqueness”

problem.• This means that there is more than one possible

geological model consistent with the seismic data.• The only way to decide between the possibilities is to use

other information, not present in the seismic data.• This other information is usually provided in two ways:

• the initial guess model• constraints on how far the final result may deviate

from the initial guess• This means that the final result always depends on the

“other information” as well as the seismic data. This is shown in the next slide.

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A solution for non-uniqueness

The flowchart above shows the general procedure for inversion. In the next few slides, we will look at building the model.

SeismicData

GeologicalConstraints

OptimumSection

GeologicalModel

Combineand

Invert

FinalInversion

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The initial guess modelThe initial guess model consists of an impedance log, which must be measured in 2-way travel time. Since the original logs are measured in depth, a critical step is depth-to-time conversion. The depth-to-time conversion is made using a depth-time table which maps each depth to the two-way travel time from the datum (surface) to that depth.

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The depth-time table is usually calculated from the sonic log velocities by:

where: tj = time down to layer j

dj = thickness of layer j

Vj = velocity of layer j

Note: The time to an event depends on all the velocities above that layer, including the first velocity to the surface, V1. That velocity is unknown and is usually approximated by extrapolating the first measured velocity back to the surface:

i

1 = j j

ji V

d2 =t

The initial guess model

2-40

If the well is deviated, it must be corrected to vertical and the correction made from KB to datum:

Dm = Measured depth from KBDv = Vertical depth from KBDs = Vertical depth from datumT = Two-way time from datum


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The depth-time table calculated from the sonic log is rarely sufficient to produce a model impedance which ties the seismic data properly because:

• The seismic datum and log datum may be different.• The average first layer velocity is not known.• Errors in the sonic log velocities produce cumulative errors in the

calculated travel-times.• The events on the seismic data may be mispositioned due to

migration errors.• The seismic data may be subject to time stretch caused by

frequency-dependent absorption and short-period multiples.

To improve the depth-time table two procedures are used:• Apply check shot corrections.• Apply manual log correlation to the seismic data


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Check shot correctionsThe depth-time table calculated from the sonic log must be modified to reflect the desired check shot times:

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(1) Change all the velocities in the log in such a way that the new log will integrate to exactly the desired times.

(2) Change the velocities for layers between the first and last check shot depth only.

(3) Do not change the velocities in the sonic log and use the depth-time table for the conversion from depth to time.

Check shot corrections - options

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Log correlation

Log correlation is the process of applying a manual correction to the depth-time curve to optimize the correlation between initial model and seismic data. Correlation consists of selecting events on the synthetic trace and the corresponding events on the real trace. The choice of wavelet is crucial.

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(a) The picks used for the correlation between the synthetic and the seismic data.

(b) The correlated seismic after stretching and squeezing the log.

Log correlation

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The correlation of the seismic data and the synthetic from the previous correlation. The symmetry of the correlation shows that the zero-phase wavelet is correct. The correlation coefficient is 0.86

Log correlation

2-47

(a) The equivalent check-shot correction for the preceding picks using a linear interpolation.

(b) The equivalent check-shot correction using a spline interpolation.

Log Correlation

2-48

Interpolating the logs

An impedance model can be built by stretching and squeezing the sonic and density logs laterally across the seismic volume. Picking two or more events is equivalent to applying a variable check-shot at each trace. The material between the two picked events is stretched/squeezed.

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When more than one well is entered into the model, the results are interpolated using inverse-distance weighting. Using picked events with multiple logs forces the inverse distance interpolation to be guided by the picked events.

Interpolating the logs

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Inversion Schemes

• Now that we have discussed the convolutional model and the initial guess impedance model, we will discuss the various inversion algorithms.

• We will start with bandlimited recursive inversion.

• We will then briefly discuss the new coloured inversion method.

• Next, we will discuss model-based inversion.

• Finally, we will discuss sparse-spike inversion.

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Now that we have discussed the general procedure for computing a model, let’s see how we can incorporate the model using different inversion schemes. We will start with the recursive inversion approach, shown in the next slide. Incorporating the model involves the following steps:

(1)Deconvolve the data using the extracted wavelet.(2)Scale the seismic trace to true reflectivity.(3)Recursively invert each trace, using the formula described earlier.(4)Add in the low frequency component of the model.

Recursive, bandlimited inversion

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Sonic/Density Logs

Scaled Seismic Data

Recursively Invert

Low Pass Filter

Add

Bandlimited Inversion

Recursive inversion

The above flowchart shows the general implementation of bandlimited inversion.

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Bandlimited inversion The resulting initial guess model for various settings of the high-cut frequency is shown below:

2-54

Issues in recursive inversion

• We assume that the samples in the seismic trace are actually reflection coefficients. This means that we assume that there is no seismic wavelet.• We assume that the samples are scaled properly. Reflection coefficients must be numbers between -1 and +1. Seismic samples may have any amplitude. • Since the equation is applied recursively from top to bottom of the trace, the effect of errors is cumulative. • The greatest effect of this cumulative error is in the trend or low-frequency component of the answer. This trend is so poorly defined that we remove the trend from the answer and replace it with the trend from the model. • The process is called bandlimited because the final impedance traces are defined within the same frequency band as the input seismic data.

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Scaling the result

The Convolutional Model is used as the basis for all inversion:

Trace = Wavelet * Reflectivity + Noise

In the frequency domain, this can be approximated by:

Reflectivity = Trace / Wavelet

To solve for the reflectivity, the wavelet must be known. Normally, when a wavelet is extracted, only its shape is known; not its absolute amplitude. Inversion requires that the absolute amplitude be known as well. From the equation above, if the wavelet is multiplied by 2, the resulting reflectivity will be divided by 2.

A good way to determine the scaling of the wavelet automatically is to force the root-mean-square amplitude of the initial guess synthetic to be equal to the root-mean-square amplitude of the real trace.

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Scaling the Result

Good Scaling:

Scaling too low:

Scaling too high:

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The density component

The coefficients, a and b, can be derived by a least-squares fit using all the wells in your area. Note that the equation is linearized by taking the logarithm of both sides.

25.0V23.0

baV

As an arbitrary relationship, we can use the generalized Gardner’s equation:

As we have seen, post-stack inversion can only give us the impedance, not the velocity. If a density log is not available, a common approximation is to use Gardner’s equation:

Vlnbalnln

Note that post-stack inversion is incapable of deciding whether a particular impedance change is a change in velocity, or a change in density, or both. This procedure assumes that the change is distributed between the two.

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Examples of bandlimited inversion

• In the next few slides, we shall see several example of bandlimited, recursive inversion.

• In our first example, we will use a wedge model. Although this model is simple, we know what the right answer should be and can therefore judge the effectiveness of the method.

• In the second example, we will look at a carbonate reef from Alberta.

• In the reef example, we want to see how well the method can image the porosity.

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Wedge model

A good illustration of inversion is provided by using the simple wedge model. The figure on the left shows the basic seismic model with the velocity log inserted at CDP 45. The wavelet and its amplitude spectrum are on the right.

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Wedge model

Here is the initial velocity model for the wedge inversion, created by picking the four major events and “squeezing” the velocity log. The colour bar is on the right.

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Wedge model

The above figure shows the bandlimited recursive inversion of the wedge model. The extra events seen throughout the inversion are due to the wavelet sidelobes.

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Reef example

Next, we will consider the inversion of a pinnacle reef from Alberta, Canada. We will first look at one line from a 3D survey that goes over the discovery well. The seismic section is shown above, with the synthetic seismogram inserted at the well location. The zone of interest is at 1150 ms.

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Reef example

Above is shown the initial model for the reef inversion. Notice that the velocities from the well have be stretched and squeezed laterally using three picked events. The box indicates the zone displayed in the following inversions.

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Reef Example

Here is the bandlimited recursive inversion around the zone of interest using a 10 Hz high-cut filter. Notice the reef porosity below 1160 ms.

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Reef Example

Here is the bandlimited recursive inversion around the zone of interest using a 20 Hz high-cut filter. Notice that there is more effect from the model than there was using a 10 Hz high-cut.

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Coloured Inversion

Coloured inversion is an inversion technique developed by Lancaster and Whitcombe of BP (SEG Expanded Abstracts, 2000, p 1572-1575).

In this approach, inversion is performed by designing an operator that maps the mean seismic amplitude spectrum to the mean earth acoustic impedance amplitude spectrum, with a -90o phase shift.

This operator is then applied to the seismic data.

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Time domain response of the coloured inversion operator.

Coloured Inversion

2-68Coloured inversion result showing relative impedance changes.

Coloured Inversion

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Model based inversionModel based inversion also follows from the convolutional model:

seismic = wavelet * reflectivity + noise

We assume that:

the seismic trace is known the wavelet is known the noise is uncorrelated and random

Model based inversion differs from bandlimited inversion in that the reflectivity is defined as that sequence which “fits” the data best. That is, if we can find a reflectivity which convolves with the wavelet to give a good approximation to the seismic trace, we assume that this is the right answer.

In practice, Model Based Inversion starts with an initial guess and improves on it by a series of steps.

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Synthetic Seismic

Seismic Data

Update Impedance

Inversion = Model

Model based inversion

The above flowchart shows the general implementation of model based inversion.

WaveletAcoustic Impedance

Difference

SmallError?

Yes

No

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Model Based InversionModel-based inversion minimizes an objective function of this form:

J = weight1 x (T - W*r) + weight2 x (M - H*r)where:

T = the seismic trace

W = the wavelet

r = the final reflectivity

M= the initial guess model impedance

H = the integration operator which convolveswith the final reflectivity to produce the final impedance

* = convolution

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Model Based InversionThe objective function has two parts:

Minimizing the first part, T - W*r, forces a solution which models the seismic trace.

Minimizing the second part, M - H*r, forces a solution which models the initial guess impedance using the specified block size.

These two conditions are usually incompatible. The weights, weight1 and weight2, determine how the two parts are balanced. This is called a “soft constraint”. As a further constraint, called a “hard constraint”, we can force the final answer to stay within the hard limits as shown below:

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Model based inversionThe initial guess model is blocked to a series of uniform blocks with this size. The final inversion result may change the size of the blocks, but the number of blocks is still the same. This means that some blocks get bigger and some get smaller, while the average is kept constant.

Using a small block size (2 ms) will increase the resolution, but the increased detail may be coming from the initial guess.

Using a small block size will always improve the fit between the input trace and the final synthetic trace.

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Model Based InversionThe seismic trace:

The wavelet:

The initial guess impedance:

Step 1: Block the initial guess impedance with a uniform block size.

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Model Based InversionStep 2: Form a synthetic trace by convolving the blocky impedance with the known wavelet:

Step 3: Compare the synthetic trace with the real trace.

Step 4: Modify both the amplitudes and thickness of the blocks to improve the fit:

Repeat this process through a series of iterations.

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Model Based InversionPotential problems with Model Based Inversion:(1) Sensitive dependence on the wavelet:

Inversion using the correct wavelet:

Inversion using the wrong wavelet

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Model Based InversionPotential problems with Model Based Inversion:(2) Non-uniqueness. For a given wavelet, all these results fit the trace about equally well:

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Inversion diagnostics

A good diagnostic for the quality of the final model based inversion result is the error plot, which is the difference between the seismic data and the final synthetic. A typical error plot is shown below:

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Examples of model based inversion

• In the next few slides, we shall see several example of model based inversion.

• In our first example, as in bandlimited inversion, we will use a wedge model. Since we know what the right answer should be, we can therefore judge the effectiveness of the method.


• Again, in the reef example, we want to see how well the method can image the porosity.

• We will also look at the effect of changing several of the key parameters in model based inversion.

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Wedge model

The above figure shows the model based inversion of the wedge model. The result is better than bandlimited recursive inversion, but starts to break down on the left, when thin bed tuning effects are encountered.

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This is the original model:

This model was created by deleting the horizon at the base of the wedge:

Wedge Model

Next, we will consider the effect of changing the initial model, as shown above.

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Original Model and Inversion Inversion with horizon removed

Wedge model

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Reef example

Here is the model based inversion around the zone of interest using a 6 ms block size. Notice that there is more detail than in recursive bandlimited inversion and that the boundaries are more clearly indicated.

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Reef example

Here is the model based inversion around the zone of interest using a 2 ms block size. Notice that there is more detail than in the previous inversion, but that this detail may be coming from the model rather than the seismic.

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Here is the full 3D inversion of the reef structure, where the top figure shows a structure slice averaged over the reef porosity, and the figure on the right shows the inverted result superimposed on the reef structure.

Reef example

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Sparse-spike inversion

Sparse-spike inversion attempts to recover a sparse estimate of the earth’s reflectivity from the seismic trace. There are two separate approaches in the literature for performing sparse-spike inversion, the maximum-likelihood (ML) approach (Chi, C. Y., Mendel, J. M. and Hampson, D., 1984, A computationally fast approach to maximum-likelihood deconvolution: Geophysics, 49, 550-565 ) and the Linear Programming (LP) approach (Oldenburg, D. W., Scheuer, T. and Levy, S., 1983, Recovery of the acoustic impedance from reflection seismograms: Geophysics, 48, 1318-1337). Briefly:

Maximum-likelihood sparse-spike inversion performs the sparse-spike estimate in the time domain by assuming that the reflectivity has a Poisson distribution.

Linear programming sparse-spike inversion performs the sparse-spike estimate in the frequency domain by assuming that the central frequency range is correct and the upper and lower frequencies constrain the spike series to be sparse.

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Sparse-spike inversion – ML approachMaximum-likelihood Sparse Spike Inversion assumes that the actual reflectivity can be thought of as a series of large spikes embedded in a background of small spikes:

Sparse Spike Inversion assumes that only the large spikes are meaningful.

It finds the location of the large spikes by examining the seismic trace.

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Sparse-spike inversion – ML approachSparse Spike Inversion builds up the reflectivity sequence one spike at a time. Spikes are added until the trace is modelled accurately enough. The amplitudes of the impedance blocks are determined using the model-based inversion algorithm.

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For maximum-likelihood sparse-spike inversion needs to know the following information in order to perform inversion:

Maximum Number of Spikes

This tells the algorithm the maximum number of allowable spikes per trace, and is normally the same as the total

number of samples in the window to be inverted.

Spike Detection Threshold

As each spike is added, its amplitude is compared with the average amplitude of all spikes detected so far. When the new amplitude is less than a specified fraction of the

average, the algorithm stops adding spikes.

Sparse-spike inversion – ML approach

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Examples of ML sparse-spike inversion

• In the next few slides, we shall see several example of ML sparse-spike inversion.

• In our first example, as in bandlimited inversion, we will use a wedge model. Since we know what the right answer should be, we can therefore judge the effectiveness of the method.


• Again, in the reef example, we want to see how well the method can image the porosity.

• We will also look at the effect of changing several of the key parameters in sparse-spike inversion.

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Wedge model

The above figure shows the sparse-spike inversion of the wedge model. The result is similar to model based inversion but shows some vertical “striping”. However, it has done better in the thin bed tuning region.

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Reef example

Here is the sparse-spike inversion around the zone of interest using a 100 spikes and an amplitude threshold of 5%. The result is comparable to model based inversion, but shows more vertical blocking.

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Reef example

Here is the sparse-spike inversion around the zone of interest using a 100 spikes and an amplitude threshold of 15%. The result is definitely less realistic than the previous result using a threshold of 5%.

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The linear programming (LP) approach to sparse-spike inversion assumes that the amplitude spectrum of the seismic trace can be divided as follows (figure is from “Sparse Spike Inversion and the Resolution Limit”, given by Qing Li at the 2002 CSEG Convention):

Sparse-spike inversion – LP approach

We first perform an optimal deconvolution to recover the seismic band of frequencies. We then constrain the output trace to be as sparse as possible, so that we recover both the high and low frequency bands. The solution is done using an L1 norm, and is implemented with a linear programming algorithm.

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Sparse-spike inversion – LP approachHere is an example from the paper by Oldenburg et al (1983):

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For linear programming sparse-spike inversion the following parameters are important:

Sparseness Ranging from 100% (as sparse as possible) to 0% (as dense as

possible.

Window length The speed of calculation can be improved by shortening the

window length used to compute frequency domain constraints. Of course, this will also reduce the accuracy of the inversion.

Maximum constraint frequency

Since no algorithm can recover the low band signal reliably (as discussed earlier) this parameter brings in model values up to this maximum frequency.

We will now look at some examples, using the above parameters.


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We will start with a series of tests using the blocky model impedance shown above. This figure and the ones that follow are taken from “Sparse Spike Inversion and the Resolution Limit”, given by Qing Li at the 2002 CSEG Convention


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The bottom curve is the initial model. The black trace in the middle is the input seismic and the red trace is the synthetic trace calculated with the initial model.


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The model displayed in red at the bottom of the figure is the inverted impedance model. The synthetic trace calculated with the inverted model matches with the input seismic trace perfectly. It yields a very small residual error, which is displayed at the top of the figure.


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We will now use a different starting model, as shown above. The bottom curve is the initial model. The black trace in the middle is the input seismic and the red trace is the synthetic trace calculated with the initial model.

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The model displayed in red at the bottom of the figure is the inverted impedance model. The synthetic trace calculated with the inverted model matches with the input seismic trace perfectly. It yields a very small residual error, which is displayed at the top of the figure.

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The figure above shows a real data example. The initial model at the bottom is extracted from nearby well log data. The input seismic trace displayed as a black curve in the middle is extracted from a post-stack seismic volume. The red trace is a synthetic trace calculated from the initial model.

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Figure 8 shows the inversion result with 100% sparseness, a 256 sample window length, and a 5 Hz maximum constraint frequency. The inverted model shows blocky characteristics. The synthetic seismic trace calculated with this model matches well with the real seismic data. In the following tests, the parameters are the same as above except when noted.

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Figure 8 shows the inversion result using a sparseness of 50%. The inverted model shows less blocky characteristics than the inversion with 100% sparseness. The synthetic seismic trace calculated with this model still matches well with the real seismic data.

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Figure 8 shows the inversion result using a sparseness of 20%. The inverted model shows less blocky characteristics than the inversion with 100% or 50% sparseness. However, the synthetic seismic trace calculated with this model still matches well with the real seismic data.

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Figure 8 shows the inversion result using a window length of 128 samples. The inverted model does not match the seismic as well as the inversion using a 256 sample window length.

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Figure 8 shows the inversion result using a window length of 56 samples. The inverted model does not match the seismic as well as the inversion using either a 256 sample or 128 sample window length.

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Figure 8 shows the inversion result using a maximum constraint frequency of 5 Hz. Notice the low frequency error that has been introduced into the error, since we ignored more of the low frequencies in the seismic data.

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Sparse-spike inversion summarySparse Spike Inversion

Puts events only where the seismic demands.

Attempts to produce the simplest possible model consistent with the data.

Often produces fewer events than are known to be geologically true.Less dependent on initial-guess model

Model-based Inversion

Puts events where the initial guess model (user) demands.

Produces the closest model to the initial guess, which is also consistent with the seismic data.

Can produce higher-resolution results than supported by seismic alone.Subject to non-uniqueness. Dependent on initial-guess model.

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A North Sea example

• In the next slide, we will consider an inversion example of a taken from a field in the North Sea.

• This example consists of a fluvial clastic anomaly.

• The model based inversion result shown here gives a good indication of the lateral extent of the sand body, especially when shown in colour superimposed on the sand structure.

2-111Average velocity over sand event.

V ft/s

North Sea Inversion Example

2-112Velocity draped on sand structure.

V ft/s

North Sea Inversion Example

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Conclusions• This has been a overview of the post-stack

inversion approach and three specific methods which are used for the inversion itself.

• We first discussed the common factors in all three methods, which involved getting a good wavelet estimate and building an accurate model.

• We then considered the simplest method, bandlimited recursive inversion, with examples.

• After this, we looked at model based inversion, and looked at examples.

• Finally, we discussed sparse-spike inversion, and looked at examples.

• In general, the model based approach produces the most geologically consistent results.

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(A) The P and S-impedances for the three cases are:

ZPgas= 3900 m/s*g/cc ZSgas= 2555

ZPwet= 5275 ZSwet= 2638

ZPshale= 4500 ZSshale= 2250

(B) The P and S reflection coefficients for shale over sand for the gas and wet cases are:

RPgas= -0.071 RSgas= 0.063

RPwet= 0.079 RSwet= 0.079

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Exercise 2-1 Answers(C) The gas and wet impedances for the P and S-wave

cases are:

39000.071 10.071- 14500

R- 1R+ 1

Z= ZPgas

PgasPshalePgas

52750.079 10.079 14500

R- 1R+ 1Z= Z

Pwet

PwetPshalePwet

25530.063 10.063 12250

R- 1R+ 1

Z= ZSgas

SgasSshaleSgas

26380.079 10.079 12250

R- 1R+ 1Z= Z

Swet

SwetSshaleSwet

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Note that we can only recover the true value for the change in impedance if we have a single spike, which is not the case after convolution with the wavelet.

V2 = 818 m/s

V2 = 1500 m/s

V1 = 1000 m/s

V3 = 1227 m/s

V4 = 1004 m/s

V1 = 1000 m/s

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Exercise 2-3: Answer


1000 1250 1500750 1750


t1

t2

Z1 = 1223

Z2 = 1000

Z3 = 1500

Reflectivity

0 0.1 0.2-0.1-0.2

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The result looks like this:




Exercise 2-4 Answer

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Exercise 2.5 Answer

Note that the resulting trace is “tuned”, meaning that the two zero-phase wavelets have been merged into a single ninety-degree wavelet.

2.05.04.0

1.0

2.01.0

102112

01


t

0.5

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Deconvolution and Inversion

We can deconvolve the seismic trace exactly if we know the wavelet matrix exactly (that is, the wavelet values and the positions of the reflection coefficients). This can be done using the generalized inverse solution r = (WTW)-1WTr, as shown below:

2.01.0

2.16.0

6/1006/1

2.16.0

6006

r

2.04.02.0

1.02.0

1.0

121000000121

102010

010201

121000000121

r

1

1

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Exercise 2-6Deconvolve the seismic trace from the previous exercise, and see if the correct reflectivity can be extracted.

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Exercise 2-6 solution

Again, we can deconvolve the seismic trace exactly if we know the wavelet matrix exactly (that is, the wavelet values and the positions of the reflection coefficients).

2.01.0

6.14.1

6446

201

6.14.1

6446

r

2.05.04.0

1.0

12100121

102112

01

12100121

r

1

1

2 Post Stack Inversion

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