seismic processing 3

Seismic Processing

77m

frequency

velocity• wavelength =

• f = 35Hz, v = 2700m/s

Any continuing constant cyclic phenomenon may be represented graphically by a sine or cosine wave. Each circle can be described by the amplitude and the wavelength.

The amplitude is the peak value to which each cycle rises or falls in reference to a base line.

The wavelength is the distance from peak to peak of one full cycle. Wavelength measured in time is period of oscillation (frequency = no. of cycles per second).

Phase is the point in the cycle at which the sequence started

These three coordinates are sufficient to describe each frequency component.

Seismic Waves

Complex signals are composed of the sum of a number of elementarywave forms, each at some constant amplitude and frequency, starting atsome defined phase angle.

Two frequencies – a base frequency and a harmonic twice the base canbe added together to produce a signal different in appearance from either of the components.

Changing the amplitude of either of the components will immediately affect the appearance of the output.

See handouts for sketches

Information Theory I

Addition of more frequencies will produce still more complex signals. The sum of frequencies representing two, four and eight periodic cycles over some interval will itself be periodic, repeating the output waveform twice over the interval.

For frequencies of 2, 4, 8 Hz, periodicity is 0.5 sec. ( 2 times in one sec and once is 0.5 sec.)

The output from the summation of a number of periodic components will itself be periodic over the interval into which all of the component wavelengths will divide evenly

In addition to variation in the amplitude and frequency, the phase of each component may vary.

Information Theory II

Fourier Synthesis – sum of zero phase monofrequencies

Fourier Synthesis – Sum of 90o phase shifted monofrequencies

Any composite signal may be changed substantially in appearance by a relatively minor change in the phase of the components

The change in phase does not change the periodicity of the signal. The peak may shift in time but the relative time difference (periodicity) will not.

Any continuous signal can be approximated by simply summing a set of individual frequency components at some specified amplitude and phase relationship.

See sketch

The complete specification of the range of frequencies contained in a given signal, defined in terms of amplitude and phase of each frequency, forms the Fourier Transform of that signal. It is a description of the signal in the frequency domain.

The computation is reversible.

Information Theory III

A time domain and frequency domain description of the same signal are known as the transform pair. Observations:

1. Frequencies near 30Hz are most dominant.2. Absence of frequencies below 5Hz (low cut filter).3. The band of energy between 5 and 10Hz has high amplitude.4. Loss of high frequencies with increasing reflection time. (What is the

impact of high frequencies).5. Individual component signals visible before the time break but flat on the

summed output.

The ability to separate a signal into its component parts is a powerful tool for analysis and operation.

The Fourier transform is completely linear. An operation in one domain has an equivalent in the other

One and only one value of amplitude and phase defines each frequency Component in the transform of the seismic trace.

Information Theory IV

Vector Analysis Each frequency component may be considered as a vector of length equal to the amplitude and directed to the corresponding phase angle.

The sum of two vectors will produce the amplitude and phase of the output frequency component.

This applies to any number of components.

Advantages of Fourier Transform

1. Quantitative analyses and comparisons of several related signals.2. Modification of signals by removal or change of specified components of the signal in a controlled manner.3. Determination of changes produced in a signal as it passes through a given process.

Information Theory V

Increasing the frequency components (adding higher frequencies to the Fourier synthesis) gives a compressed signal in the time domain if the zero phase character is unchanged.

If all the frequencies in the inverse Fourier transformation are included, then the resulting wavelet becomes a spike. Therefore, a spike is characterized as the in-phase synthesis of all frequencies from zero to the Nyquist.

For all frequencies, the amplitude spectrum is of a spike in unity, while its phase spectrum is zero.

Pictures.

Information Theory VI

Common Relationships A sinusoidal wave can be designated by a single point having coordinates of amplitude and frequency on the amplitude spectrum.

A single spike in the time domain contains all frequencies.

True random noise measured over a sufficiently long time period will also contain equal amounts of all frequencies.

What is the difference between the two?

Information Theory VII

Once a continuous function is sampled all information between the samples is lost.

Therefore the signal must be sampled often enough to make sure the set of samples truly represent the signal. The sampling interval is the rate in time at which the signal is sampled. Typical values vary between 1 – 4ms

A fundamental rule:

Two cycles per second must be taken if any given frequency is to be defined properly. The sampling rate must be at least twice the frequency to be sampled. This sets a mathematical limit of the highest frequency to be sampled.

Highest frequency which can be defined correctly is Nyquist frequency

Frequencies higher than Nyquist frequency corresponding to a given sample rate will appear on the frequency spectrum having frequency equal to diff between signal frequency and the sampling frequency. Signal frequencies outside the limit fold back into the frequency sprectrum .

Sampling Theory I

Filters are designed to remove any frequency above the Nyquist frequency .

If sampling rate is t the Nyquist freq. is 1/(2 t)

If t = 2ms Nq = 250Hz

What of 4ms, 8ms

The coarser the sampling interval, the smoother the signal (resulting from the loss of higher frequencies).

If the adequate sampling interval is not used the higher frequencies in the seismic trace will be lost.

Frequencies above the Nyquist frequency fold back into the spectrum (aliasing)

Formula for aliased frequency.

Sampling Theory II

Phase Considerations – Zero Phase

Inverse Fourier Transform

Zero phaseSummation gives a time domain signal called a wavelet. A wavelet is a transient signal i.e. a signal with a finite duration. It has a start time and a finish time and its energy is confined withinthe two time positions.

The wavelet is symmetrical around time t = 0 and has a positive peak amplitude at t = 0. Such a wavelet is called zero phase.

The wavelet was synthesised using the zero-phase sinusoids of equal peak amplitude.

Phase Considerations – Time Shift

A zero phase wavelet is symmetrical with respect to zero time and peaks at zero time. The wavelet has shifted by – 0.2secs. Its shape has not changed.

A linear phase shift is a constant time shift applied to the wavelet.

Phase Considerations – Time Shift

Phase Considerations – Phase Shift

If 90 degree is applied to each of the sinusoids and the zero crossing aligned at t = 0, then and anti symmetrical wavelet is derived.

They wavelets all have the same amplitude spectrum. The difference in wavelet shape is due to difference in phase spectra.

Phase Considerations – Phase Shift

Phase Considerations – Phase + Time Shift

A time shift combined with a phase shift results in a time shifted antisymmetrical wavelet.

By keeping the amplitude spectrum unchanged, the wavelet shape can be changed by modifying the phase spectrum.

Phase Considerations – Modifications to Phase

Bandwidth and Vertical Resolution

All frequencies are needed; not just high frequencies alone.

Seismic Processing Domains

CMPShot Stack Migration

Shot Domain

Offset

Time

CMP domainshot domain stack domain migration domain

Resorting from Shot Domain to CMP Domain

Sort:

CMP

CMP

Time

Fold = 4


Stack Domain

Apply NMO

Stack

Time

Offset

Time

Offset CMP

Time

CMP


Seismic Processing

Velocity Analysis/NMO

A shot record is the collection of seismic traces generated when one source shoots into many receivers. Dots below the reflector show subsurface reflection points (halfway between the source and a receiver – midpoint). These are shown as black dots above the acquisition surface As the offset increases, so does the travel time from source to receiver. This characteristic delay of reflection times with increasing offset is called normal moveout.

NMO Basics - I

A seismic processing step whereby reflection events are flattened in a common midpoint gather in preparation for stacking. As the shots roll along, there will be many source-receiver pairs with the same CMP location. The reason for gathering multifold data is that we get redundant information which can be used to reduce noise and create a more reliable image. Our goal is to eventually process all these traces as a family and add them together (CMP stack) to make one trace that lives at this CMP location.

NMO is aimed at removing thehyperbolic curvature in reflectionEvents i.e.removing the effect ofoffset. The reflection shouldcome in at the same time for alloffsets (since we have removedany travel time delay due tooffset). In short, reflection eventsshould be flat after NMO

NMO Basics – What is NMO?

Reflections can be seen in real data along with other kinds of events. There are receivers on both sides of the shot shown. The right side has been marked-up to identify different kinds of events – direct arrivals (p-wave, s-wave, air wave, surface wave), head waves and (a few) reflections. The left side is uninterpreted. The reflection events have a hyperbolic shape characteristic of normal moveout.

NMO Basics - II

The time difference between traveltime at a give offset and at zero offset is called normal move out (NMO)

The velocity required to correct for NMO is called NMO velocity.

NMO CorrectionUsing the correct NMO velocity in very important. If a higher velocity is used, then the hyperbola is not completely flattened. This is undercorrection. If a lower velocity is used, overcorrection results.

Usually NMO correction is applied to the input CMP gather using a number of trial Constant velocity values (panels). The velocity that best flattens the reflection hyperbola is the velocity that best corrects fro NMO before stacking the traces in the gather

NMO Basics - III


t(x) = traveltime along raypathx = offsetv = velocity of medium abovet(0) = travel time along vertical path

NMO Correction

Appropriate velocity2264m/s

OvercorrectionToo low velocity2000m/s

UndercorrectionToo high velocity2500m/s

The layers have interval velocities (v1, v2, …..vN) where N is the number of layers.

t2(x) = Co + C1x2 +C2x4 + C3x6 + ………..

Co = t2(0), C1 = 1/v2rms and C2, C3

Are complicated functions that depend on layer thicknesses and interval velocities.The rms velocity down to the reflector on which depth point D is situated is defined as

= vertical 2 way travel time through the ith layer and

By making the small spread approximation (offset small compared to depth), the series in the equation can be truncated as follows

t2(x) = t2(0) + x2/v2rms

The velocity required for NMO correction for horizontally stratified medium is equal to the rms velocity, provided the small spread approximation is made

)0()0(

1

1

21

2i

N

irms tv

tv

it

i

kktt

1

)0(

NMO in a horizontally stratified earth

NMO in a horizontally stratified earth

t2(x) = t2(0) + x2/v2NMO + higher order terms

Final equation:t2

st(x) = t2st(0) + x2/v2

st

NMO for several layers with arbitrary dips

As a result of NMO correction a frequency distortion occurs, particularly for shallowevents at large offsets. This is called NMO Stretching.

The waveform with dominant period T is stretched so that its period T’ after NMO Correction is greater than T.

Stretching is a frequency distortion in which events are shifted to lower frequencies.

Stretching is quantified as f/f = tNMO/t(0)

The stretching is normally confined to large offset and shallow times.

Muting is applied to correct for the stretching.

NMO Stretching

NMO Stretching I

NMO Stretching II

Velocity Analysis

The sonic log is a direct measurement of velocity while the seismic method is an indirect method.

Based on these two types of velocity information, you can derive a large number of velocities:

• Interval (velocity in an interval b/w twp reflectors)• Apparent• Average• Root mean square (rms)• Instantaneous• Phase• Group• NMO• Stacking• Migration

The velocity that can reliably be derived from seismic is the velocity that yields thebest stack.

Velocity Analysis

Interval velocity is the average velocity in a interval between two reflectors.

Factors affecting interval velocity:

• Pore shape• Pore pressure• Pore fluid saturation• Confining pressure• Temperature

Velocity increases with confining pressure – i.e. with depth (the most important)Possibility of velocity inversion due to pore pressure.

Velocity Analysis

Vs

Vp

1

5

3

S

D

S

D

0.8 2.4 1.6

Vs

Vp

2

3

4

0.2 0.6

Confining Pressure (kbar)

Vel

ocit

y (k

m/s

)

Confining Pressure (kbar)

Vel

ocit

y (k

m/s

)

• Velocity increases rapidly with confining pressure at small confining pressures, then gradually levels off• P wave velocity is greater that S wave velocity regardless on confining pressure • The saturated rock sample (S) has a higher P-wave velocity that the dry (D) sample – why?• At higher confining pressures the saturated and dry samples have the same P-wave velocity• P-wave velocity in the saturated sample does not change as rapidly as the dry sample• Fluids do not support S-waves

Pores as microcracks

Rounded pores

Velocity Analysis

t2-x2 method - from the equation:

t2st(x) = t2

st(0) + x2/v2st

A practical way of determining the stacking velocity from the CMP gather can be determined. the equation describes a line on the t2(x) versus x2 plane.

The slope is 1/v2st and intercept at x = 0 is t(0)

Claerbout’s method• Measure slope along a slanted path that is tangential to both the top and bottom reflections (Slope 1).• Connect the two tangential points and measure the slope of this line (Slope2).• The interval velocity is the square root of the product of the two slopes values.

Velocity Analysis II

Velocity Analysis III

Real data example

Velocity Analysis

Constant Velocity Scans• Display NMO corrected gathers from different velocities in panels.• The most reliable velocity gives the best stack.• Stacking velocities are estimated from data stacked with the range of constant velocities on the basis of stacked event amplitude and continuity.

Velocity spectrumMove from the offset vs. two way time domain to the stacking velocity vs. two way zero offset time domain.

Velocity Analysis IV

Velocity Analysis I

Velocity Analysis

Factors Affecting Velocity Estimates:

• Spread Length• Stacking fold• S/N ratio• Muting• Time Gate Length• Velocity Sampling• Choice of Coherency Measure• Time departures from hyperbolic movement• Bandwidth of Data

Velocity Analysis

Spread Length:Adequate resolution in the velocity spectrum can only be obtained with a sufficiently large spread that spans both near and far offsets. Lack of large offset means lack of significant moveout required for velocity discrimination.

Stacking fold:The lower the stacking fold the lower the resolution of velocity analysis

S/N ratio:Noise on seismic data has a direct effect on the quality of a velocity spectrum. The accuracy of the velocity spectrum is limited when the S/N ratio is poor.

Muting:Muting reduced fold for shallow data and has an adverse effect on the velocity spectrum

Time Gate LengthIf the gate is chosen too small, computational costs increase. If too coarse the spectrum suffers from lack of temporal resolution. The gate length is chosen between one-half and one times the dominant period of the signal, typically 20 to 40ms.

Influence of Maximum Offset

Influence of Random Noise

Velocity Analysis

Velocity SamplingVelocity range should correspond to those velocities to those of primary reflections present on the CMP gather. Velocity increment must not be too coarse.

Choice of Coherency MeasureCompare different gathers.

True departures from hyperbolic moveoutSpecial correction is required.

Bandwidth of DataChoose a wide corridor to cover velocity variations vertically and laterally in the survey area.

For horizontal layers, CDP = CMP. For a dipping layer the two are not the same.

t2(x) = t2(0) + x2cos2 /v2rms

The NMO velocity now is given by the medium velocity divided by the cosine of the dip angle.

vNMO = v/cos

Proper stacking of a dipping event requires a velocity that is greater than the velocity of the medium above the reflector.

NMO for a dipping layer

Why stacking?

To improve S/N ratio Obtain zero offset / normal incidence trace Data reduction Attenuate multiples Obtain velocity information


Multiples

Attenuation of multiples by stacking

Seismic Processing Objectives

Data identing and editing Noise reduction Correction for elevation, source depth, shallow

anomalies Compensation of loss of amplitude Compensation for loss of bandwidth Multiple removal Imaging Wavelet processing

Wavenumber

Temporal frequency is no. of cycles per sec. The fourier dual is spatial frequency which is no. of cycles per unit

distance or wavenumber. In a dipping event, count the number of peaks within a unit distance

say 1km. along the horizontal direction. Nyquist wavenumber is [1/(2* trace interval)] = 20 cycles/km since

trace spacing is 25m. Compute wavenumber in section on next page.

• Compute total time dip across section – (23traces/section) x (15ms/traces) = 345 ms/section

• Convert this to cycles by dividing by the temporal period:• (345ms/section)/[(1000ms/s)/(12 cycles/sec)] = (4.14 cycles/sec)• Spatial extent is 575m therefore, wavenumber is:• (4.14cycles/sec)/(0.575km/section) = 7.2 cycles/km.

Wavenumber - the f/k plane

6 gathers containing 6 Hz monofrequency events with dips ranging from 0 to 15ms/trace. Trace spacing is 25m. Bottom row: Their respective amplitude spectra. Dots represent mapping of events on the gathers. +ve dips are defined as downdip from left to right so all events map to the +ve quandrant. Zero dip is equivalent to zero wavenumber.

The f-k plane corresponds to the x-t plane in the time domain. Compute inverse of stepout dt/dx I.e dx/dt.

575m/0.345s = 1.67km/sf/k = (12 cycles/s)/(7.2 cycles/km) 1.67 km/s

The higher the dips the higher the wavenumbersSpatial aliasing.The higher the frequency the smaller the dip at which aliasing occurs.


24Hz

48Hz36Hz

60Hz 72Hz


Same as earlier pictures except using 60 Hz and 72Hz.


Six gathers, each formed by summing gathers of the like dips in earlier pictures (i.e. similar dips but different frequencies). Trace spacing is 25m. Amplitude spectra shown in bottom row.

Single event, no frequency is spatially aliased

Single event, frequencies beyond 21Hz are spatially aliased



6 events, no frequencies are spatially aliased

6 events, frequencies beyond 21Hz are spatially aliased

A single isolated event sampled at three different trace spacings. No spatial aliasing occurs with the 12.5m spacing.

How do you avoid spatial aliasing?

• Apply time shifts so that steeper events appear to have lower dips.

• Low pass filter to retain required low frequencies

• Use smaller trace spacings.


Wavenumber - f/k plane

Wavenumber – dip filtering

A signal with three frequency components A, B, and C, sampled at three different rates, 2, 4, and 8 ms. Frequency aliasing occurs at coarser sampling interevals.


This is a high resolution land shot record. Some initial corrections have bben applied to the data (including statics), and the panel on the right shows the part of the data highlighted in (very light) red. .


The offsets in the above section increment by 25 m, giving a spatial range of (500/25) 20 cycles per kilometre. Here then is the FK analysis of the above shot, temporal frequencies from bottom to top, and spatial frequencies across the top. The amplitude scale is in dB, from white through blue to yellow and red. You can see the (temporal) band limited nature of this data (from about 10 to 90 Hz), some strong dips (probably the first breaks) and even some spatial aliasing (the yellow line ending at the "W" of "Wavenumber


The same FK analysis after a heavy "dip" or "fan" filter. We have removed all events with dips in excess of 2.6 ms per trace (a horizontal velocity of 25/0.0026 = 9615 m/s)

Here's what the above filter looks like in three dimensions - a wedge shaped filter (these filters are sometimes called "pie-slices") with sharp edges. In practice we would normally smooth the edges a little to avoid sharp changes in the frequency domain - these introduce long anomalies in the time domain.


The inverse transform of the filtered FK spectrum.

We have removed all of the high dips (especially the ground roll), but the data now has a slightly "wormy" appearance (it lacks spatial detail) indicating that we've probably overdone the spatial filtering.

One common way of reducing the effects of a heavy FK filter is to "mix-back" some of the original (unfiltered) data with the output. We are generally allowed to make this mix-back time and space variant, which allows us to modify the filter response for different parts of the record.

We determine this, as usual, by testing

Pre-processing

Improving signal-to-noise ratio

Improving vertical resolution

Improving horizontal resolution

Post-processing

Field data

Geometry definitionediting amplitude corrections

Static corrections

F-K filtering

Sorting to CMP domain

Velocity analysis, dynamic (NMO) corrections, stacking

Deconvolution

DMO

Migration

Filter

Display

TOPO

Improving signal-to-noise ratio

Improving horizontal resolution

Generalised Processing Sequence

Seismic Processing Step-by-Step:Identing


Shot Domain Processing(signal processing)

De-absorption Statics Deconvolution Ground Roll Removal Source & Survey Matching Zero-Phasing


Static Correction: Shot – Receiver Statics

Datum

Shot

Geophone

Shot static Receiver static

Weathering layer,Low velocity Layer (LVL)


First Break Statics


Velocity of weathering layer Elevation of weathering layer

Source statics Receiver statics

M. Bevaart, NAM

Successful Statics


Stack with simple statics

Stack with full range of statics

Seismic Processing

Deconvolution

A pure signal having some well defined shape or characteristic is transmitted from a source and is later received, contaminated by noise, at some distant recording point.

The problem – That of retrieval

The fundamental concept of seismic exploration is to send into the earth a short signal which is then reflected back from a boundary between two units.

The impulse response of a system is the output signal when a spike is the input signal.

Signal Theory

The simple basic model of the system includes a single spike or impulse input which is modified by the system to a different form at the output.

The effect of the system on the input might be described as a stretching in time and a change of shape.

The input is a spike which all values as zero except at one

An integration of all frequencies is a spike at zero phase

The System Model I

The signal from the output is not at all like the input but extends over several samples and is not symmetrical.

Signals that are not symmetrical are not at zero phase. Some phase shifting must have occurred in the transmission of the signal through the system.

The output of the system is a waveform different from the input.

The transmission medium, earth acts as a filter to remove all but a very limited band of frequencies from the broad band input.

The combined effect of the loss of several of the input frequency components and the phase shifting of the remainder produces a wavelet which is quite different from the sharp source spike. The result is defined as the unit impulse response.

The System Model II

The seismic signal travels from the source to the geophone through the transmission medium, the earth, which acts as a filter to remove all but a very limited band of frequencies from the broad band input.

The amplitude and phase response define the changes that have taken place to the input which would have amplitude at 100% for all frequencies at the zero phase.

The time domain response to a unit impulse is the wavelet.

The System Model III

Consider:1. A reflectivity sequence (1, 0, ½,)2. Impulsive source with explosion at t = 0 with amplitude 1

The response of the reflectivity sequence to the impulse is the impulse response of the system.

Time of onset ReflectivitySequence Source Response

0 1, 0, ½ 1 1, 0 , ½

One unit time later, the impulsive source generates an implosion with amplitude -½


0 1, 0, ½ -½ -½, 0 , -¼

Time Domain Operations

Since a general source function is considered to be sequence of explosive and implosive impulses, the individual impulse responses are added to obtain the combined response. This process is called linear superposition.


0 1, 0, ½ 1 1 0 ½ 1 1, 0, ½ -½ -½ 0 -¼Superposition 1 -½ 1 -½ ½ -¼

i.e (1, 0, ½ )*(1, -½) = (1, -½, ½, -¼)

* Is called convolution

Time Domain Operations- Superposition

Convolution of a source wavelet (1, -½) with the reflectivity sequence (1, 0, ½ )

Reflectivity OutputSequence Response

1 0 ½ -½ 1 1 -½ 1 -½

-½ 1 ½ -½ 1 ½ -¼

Convolution of a source wavelet (1, 0, ½ ) with the reflectivity sequence (1, -½)Reflectivity OutputSequence Response

1 -½ ½ 0 1 1 ½ 0 1 -½

½ 0 1 ½ ½ 0 1 -¼

Time Domain Operations - Convolution

1 3 1 2 3 1 1

3 2 1x x x

Convolution

Output

Measurement of similarity or time alignment of two traces is sometimes required in processing. Correlation is used to make such measurements. Consider two wavelets:

1. (2, 1, -1 , 0, 0) 2. (0, 0, 2, 1, -1)

Identical but wavelet 2 is shifted by two samples wrt. wavelet 1. The time lag at which they are most similar can be determined. Carry out what was done for convolution on wavelet 1 without reversing wavelet 2.

2 1 -1 0 0 Output Lag 0 0 2 1 -1 -2 -4 0 0 2 1 -1 1 -3 0 0 2 1 -1 6 -2 0 0 2 1 -1 1 -1 0 0 2 1 -1 -2 0 0 0 2 1 -1 0 1 0 0 2 1 -1 0 2 0 0 2 1 -1 0 3 0 0 2 1 -1 0 4Most identical at time lag –2 (6)

Time Domain Operations – Crosscorrelation

Interchange the two arrays: 1. (2, 1, -1 , 0, 0) 2. (0, 0, 2, 1, -1)

0 0 2 1 -1 Output Lag 2 1 -1 0 0 0 -4 2 1 -1 0 0 0 -3 2 1 -1 0 0 0 -2 2 1 -1 0 0 0 -1 2 1 -1 0 0 -2 0 2 1 -1 0 0 1 1 2 1 -1 0 0 6 2 2 1 -1 0 0 1 3 2 1 -1 0 0 2 4Most identical at time lag 2 (6). If wavelet 2 were shifted forward 2 samples the 2 wavelets will be maximally similar.

Unlike convolution, crosscorellation is not commutative: I.e the output depends on which array is fixed and which is moving.

Time Domain Operations – Crosscorrelation

1 6 3

3 2 6

1 4 9

2 6 3

3 2 3

1 2 0

1 0 0

1 3 1 2 3 1 1

1 2 3x x x

10

11

14

11

8

3

1

Cross Correlation

Output

Crosscorrelation of a time series with itself is called autocorrelation

2 1 -1 0 0 Output Lag 2 1 -1 0 0 0 -4 2 1 -1 0 0 0 -3 2 1 -1 0 0 -2 -2 2 1 -1 0 0 1 -1 2 1 -1 0 0 6 0 2 1 -1 0 0 1 1 2 1 -1 0 0 -2 2 2 1 -1 0 0 0 3 2 1 -1 0 0 0 4

Maximum correlation occurs at zero lag. The output is also symmetrical.

Time Domain Operations – Autocorrelation

1. Convolution is multiplication2. Phases are additive in convolution and subtractive in correlation.

A zero phase band limited wavelet can used to filter a seismic trace. The output contains only those frequencies present in the wavelet. This is called zero phase frequency filtering.

Multiply the amplitude spectrum of the seismic trace by that of the filter operator (convolution in the time domain).

Frequency Domain Operations

Spike

Frequency Domain Operations

Construct a zero phase wavelet with an amplitude spectrum that meets one of these:

Band pass (to eliminate groundroll and high frequency ambient noise) Band rejectHigh pass (low cut)Low pass (hight cut

Ground Roll (Surface Wave) Removal


M. Bevaart, NAM

Crosscorrelation of a pure sine or cosine wave with a signal will extract that same frequency component from the signal. Information regarding any other components of the signal will be lost from the output.

This is used to advantage in band pass filtering

In autocorrelation the signal is correlated with itself i.e. square each component of the amplitude spectrum and subtract the phase spectrum from itself which will yield a zero phase output.

Filtering

Time Variant Filtering

Deconvolution

Stacking can remove noise but there is still the problem of the filtering caused by the earth.

The process by which the attenuated elements are restored is called deconvolution. The impulse response of the earth contains primary reflections (reflectivity series), multiples and noise.

It is the process designed to reverse the effects of the passage of the input signal through the earth. It improves the temporal resolution of the seismic data by compressing the basic seismic wavelet.

Autocorrelate the signal to get a spike

The Convolutional Model

Sonic and density logs provide measurements of the acoustic impedances and impedance contrasts at various rock boundaries. This is used in deriving the forward model with some assumptions.

1. The earth is made up of horizontal layers of constant velocity2. The source generates a compressional plane wave that impinges on layer

boundaries at normal incidence. Under such circumstances, no shear waves are generated.

Assumption 1 fails in structurally complex areas and assumption 2 implieszero offset.

Based on 1 and 2 we have:

1122

1122

VVVVR

Generation of Synthetic Seismograms

The Reflectivity Series

a - sonic log, b - reflection coefficient series, c – b in twt domaind – impulse response, e – synthetic seismogram

The Convolutional Model (2)

The characteristic wave created by an impulsive source such as dynamite or airgun is called the signature of the source. All signatures can be described as band limited wavelets of finite duration.

As the wavelet travels through the earth, two things happen.

1. Overall amplitude decays – wavefront divergence

2. Frequencies are absorbed

At any time the wavelet is not the same as it was at the onset. This time dependent change in waveform is called non-stationarity A compensation for non-stationarity is carried out before deconvolution (spherical spreading function).

Assumption 3 The source waveform does not change as it travels in the subsurface; i.e. it is stationary.

Waveform Nonstationarity

As the wavelet travels into the earth, the amplitude level drops (geometric

spreading) ans a loss of high frequencies occurs


For reflection coefficients that are closely spaced (sparse spike series), the response to the basic wavelet is a superposition of the individual impulse responses. This is a linear process called superposition. It is achieved computationally by convolving the basic wavelet with the reflectivity series.

To identify closely spaced reflecting boundaries from the composite response, the source waveform must be removed to obtain the sparse spike series. The reverse process is called deconvolution. Deconvolution tries to recover the reflectivity series.

The building block of convolution is:

x(t) = w(t) * e(t) + n(t)

x(t) = recorded seismogram

w(t) = basic seismic wavelet

e(t) = earth’s impulse response

n(t) = random ambient noise

* = convolution

The Principle of Superposition

The Convolutional Model

Noise – a pure random series with infinite length has a flat amplitude spectrum and an autocorrelogram that is zero at all lags except the zero lag.


Deconvolution tries to restore the reflectivity series from the seismogram.

All that is known in the equation is x(t) the recorded seismogram.

The earth’s impulse response e(t) must be estimated elsewhere apart from the borehole location with good sonic logs.

The source waveform w(t) is unknown.

There is no a priori knowledge of the ambient noise n(t).

To solve for the unknown e(t), we have to make further assumptions

Assumption 4. The noise component n(t) is zero

Assumption 5. The source wavelet is known

With this we have only one unknown - e(t) in the equation. But in reality neither of these two assumptions is valid. Therefore, we examine the convolutional model further in the frequency domain to relax assumption 4.


Frequency DomainIf the source signature is known, then the solution to the convolutional model is deterministic. If not known, then the solution is statistical.

Convolution in time domain is equivalent to multiplication in the frequency domain.

There is similarity in the overall shape of the amplitude spectrum of the wavelet and the seismogram.

A smoothed version will be indistinguishable from the amplitude spectrum of the seismogram.

Assumption 6 The seismogram has the characteristics of the seismic wavelet in that their autocorrelations and amplitude spectra are the same.

This is the key to implementing the predictive deconvolution.

Frequency Domain

The Convolutional Model – Inverse Filtering (6)

If a filter operator a(t) were defined such that convolution of a(t) with the known seismogram x(t) yields an estimate of the earth’s impulse response e(t), then e(t) = a(t) * x(t)

Substitution e(t) in x(t) = w(t) * e(t):

x(t) = w(t) * a(t) * x(t) Eliminating x(t) from both sides of the equation:

(t) = w(t) * a(t) ….a

where (t) = Kronecher delta function

By solving equation …a above, for the filter operator a(t) we obtain:a(t) = (t) * w’(t)

w’(t) = inverse of the seismic wavelet w(t) which is assumed to be known

δ

δ

δ1, t = 0

0, elsewhere

(t) =

δ

δδ

The Convolutional Model – Inverse Filtering (6)

Therefore the filter operator needed to compute the earth’s impulse response from the recorded seismogram turns out to be the mathematical inverse of the seismic wavelet.

Equation ….a implies that the inverse filter converts the basic wavelet to a spike at

t = 0. Likewise, the inverse converts the seismogram to a series of spikes that defines the earth’s impulse response.

Therefore, inverse filtering is a method of deconvolution provided the source waveform is known (deterministic deconvolution).

δδ

How do we compute the inverse of the seismic wavelet?

The z transform is used to mathematically compute the inverse of the seismic wavelet - w’(t).

If the basic wavelet is a two point time series given by (1, - ), the z transform is defined by the following polynomial

W(z) = 1 – ( )z

The z transform

½

½

The power of variable z is the number of unit time delays associated with each sample in the series.

The first term has zero delay, so z is raised to zero power. The second term has unit delay, so z is raised to first power.

Hence the z transform of a time series is a polynomial in z, whose coefficients are the values of the time samples.

The inverse of the wavelet w’(t) is obtained by polynomial division of the z transform.

W’(z) = 1/ [1 – ( ½ )z ] = 1 + (½ )z + (¼)z2 + .......

The inverse time series is the coefficient of W’(z) i.e. [1, (½ ), (¼), ……]. This is the filter operator a(t). Note it has an infinite number of coefficients, although they decay rapidly. As more terms are included in the inverse filter, the output is closer to being a spike.

Convolve with more terms in the inverse filter

The z transform

Consider the two point operator [1, (½)]

Convolution of this operator with the wavelet yields [1, 0, (-¼)]. The ideal result is a zero delay spike (1,0,0). Although not ideal the actual result is spikier than the input wavelet, [1, (-½)].

1 -½ Output ½ 1 1 ½ 1 0 ½ 1 -¼

The three point filter is spikier. As more terms are included in the the inverse filter, the output is closer c;loser to being a spike at zero lag.

The z transform

1 -½ Output ¼ ½ 1 1 ¼ ½ 1 0 ¼ ½ 1 0 ¼ ½ 1 -1/8

Convolution of the truncated inverse filter [1, (½)] with the input wavelet [1, (-½)]

Convolution of the truncated inverse filter [1, (½), ¼] with the input wavelet [1, (-½)]

The inverse of the input wavelet [1, (-½ )] has coefficients that rapidly decay to zero.

What of the inverse of the input wavelet [(-½), 1)]? Here the polynomial division gives the divergent series (-2, -4, -8 ……..). Truncate this and convolve with the two point operator. The result is far from the desired output.

The inverse filter coefficients increase in time rather than decay! What happens if 8 is kept as one of the coefficients?

The z transform

-½ 1 Output -4 -2 1 -4 -2 0 -4 -2 -4

Convolution of the truncated inverse filter [(-2, -4)] with the input wavelet [(-½),1]

1 -½ Actual Desired

Output Output b a a 1 b a b - a/2 0 b a - b/2 0

Convolution of a filter (a, b) with the input wavelet [(1, -½)]


Least Square Inverse FilteringThe least square filter yields less error when converting the input wavelet to a spike.

Minimum Phase

Error in converting [1, (- ½ )] to a spike is less than that of [(- ½ ) ,1].

The first wavelet is closer to being a zero delay wavelet (1,0,0) than the second wavelet. The second wavelet is closer to being a delayed spike (0,1,0).

The error is reduced if the desired output closely resembles the energy distribution in the input series. Wavelet 1 has more energy at the onset, wavelet 2 more energy at the end.

Minimum Phase

A wavelet is minimum phase if its energy is maximally concentrated at its onset

A wavelet is maximum phase if its energy is maximally concentrated at its end

In all in-between situations the wavelet is mixed phase

A wavelet is defined as a transient waveform with a finite duration i.e. it is realizable.

A minimum phase wavelet is one-sided in that it is zero before t = 0. A wavelet that is zero for t < 0 is called causal. A minimum phase is causal and realizable.

2

1

Minimum Phase

Three wavelets with the same amplitude spectrurn, but with a different phase spectra. As a result, their shapes differ. The wavelet at the top has more energy concentrated at the onset, middle has its energy concentrated at the centre and the wavelet at the bottom has its energy concentrated at the end.

Minimum Phase

Consider the following four 3-point wavelets

A: (4, 0, -1)B: (2, 3, -2)C: (-2, 3, 2)D: (-1, 0, 4)

Compute the cummulative energy of each wavelet at any one time (add the square of the amplitudes).

0 1 2A 16 16 17B 4 13 17C 4 13 17D 1 1 17

A builds up energy rapidly, has the least energy delay while D builds up energy slowly and has the largest energy delay. A minimum phase wavelet has the least energy delay.

Minimum Phase

Amplitude spectrum for wavelets A,B,C,D (all have same amplitude spectrum)

A has the minimum phase change that is why it is referred to as minimum phase. Wavelet D has the maximum phase change across the frequencies and is referred to as maximum phase. C and B mixed phase.

Minimum Phase

The autocorrelations of the wavelets give the same output. The zero lag of the autocorrelation is equal to the total energy contained in each wavelet (i.e. 17 units). The process by which the seismic wavelet is compressed to a zero-lag spike is called spiking deconvolution.

Performance depends on filter length and whether the input wavelet is minimum phase. The spiking deconvolution operator is the inverse of the wavelet. If the wavelet were minimum phase, then we get a stable inverse which is also minimum phase.

The term stable means that the filter coefficients make a convergent series i.e. the coefficients decrease decrease in time (and vanish at time t = ∞) therefore filter has finite energy. This is the case for wavelet [1, (- ½ )] with an inverse [1, (½ ), (¼ ), ……]. The inverse is a stable spiking deconvolution filter.

If the filter were maximum phase, then it does not have a stable inverse. This is the case for the wavelet [(- ½ ) ,1], whose inverse is given by the divergent series [-2, -4, -8, …..]. A mixed phase does not have a stable inverse.

Assumption 7: The seismic wavelet is minimum phase. Therefore it has a minimum phase inverse.

Weiner filters

Filters are designed to make the output approximate a spike as much as possible.

The least squares error between the actual and desired outputs is minimum

Spiking deconvolution is when the output is a zero lag spike. It is also identical to least squares inverse filter.

Predictive deconvolution is when the output can be predicted given a particular type of input. The idea is to predict the value of the input at some particular time.

Deconvolution - Example

Undeconvolved Deconvolved0

1.0

2.0

Deconvolution –here to remove waterbottom reflections


Without deconvolution With deconvolution

P (-)2R2P-RP

Field Statics Corrections

Seismic Processing

Migration

Migration is a tool used in seismic processing to get an accurate picture of underground layers. It involves the geometric repositioning of return signals to show an event (layer boundary or other structure) where it is being hit by the seismic wave rather than where it is picked up. Migration was first used in the 1920's, and today, it is has evolved into many variations. Two of the more important migration methods are: pre-stack and post-stack migration.

Pre-stack migration is essentially when seismic data is adjusted before the stacking sequence occurs. The popular form of pre-stack migration is depth migration (PDM). PDM requires the user to know more about velocites of the layers. Once the user inputs these into the data with velocity analysis methods, there will be some error in the image. This error is caused by dipping reflectors or diffractions. The PDM will adjust the picture according to the velocities given. Pre-stack migration is often applied only when the layers being observed have complicated velocity profiles, or when the structures are just too complex to see with post-stack migration. Pre-stack is an important tool in modeling salt diapirs because of their complexity and this has immediate benefits if the resolution can pick up any hydrocarbons trapped by the diapir. Overall, pre-stack migration, depth and time, is a valuable tool in better imaging seismic data, but it is limited by the amount of time and money required to conduct a pre-stack migration. Most of the pre-stack migration will be run when post-stacking has failed to resolve the layers or structures. However, with advances in computers, pre-stack migration will eventually become more economical.

Pre Stack Migration

Post stack migration is the process of migration in which the data is migrated after it has been stacked. This process is for many reasons, mainly because of its reasonable cost compared to pre-stack migration. As in pre-stack migration, post stack migration is based on the idea that all data elements represent either primary reflections or diffractions. This is done by using an operation involving the rearrangement of seismic information so that reflections and diffractions are plotted at their true locations.

The reason that migration is needed is due to the fact that variable velocities and dipping horizons cause the data to record surface positions different from their sub-surface positions. The stacking is accomplished by making a composite record by combining traces from different records. Filtering is involved with stacking because of timing errors or wave-shape difference among the data being stacked.

A disadvantage of using post stack migration compared to pre-stack migration is that it does not give as clear results as pre-stack. Post stack usually gives good results though, when the dip is small and where events with different dips do not interfere on the migrated section.

Post Stack Migration

Migration

Post Stack Migration Pre Stack Migration

Migration moves dipping reflectors into their true subsurface positions and collapses diffraction, thereby delineating detailed subsurface features such as fault planes. In this respect migration can be viewed as a form of spatial deconvolution that increases spatial resolution.

Note that migration does not displace horizontal events, rather, it moves dipping events in the updip direction and collapses diffractions, thus enabling delineation of faults.

The goal of migration is to make the stacked section appear similar to geologic cross section across the seismic line. The migration that produces a migrated time section is called time migration.

When the lateral velocity gradients are significant, time migration does not produce the true subsurface picture. Instead depth migration is used, the output of which is a depth section.

Migration

a - CMP stack, b – Migration, c – sketch of of prominent diffraction D and dipping event before (B) migration and after (A) migration.

Migration

Migration

Migration

Depth Migration - BTime Migration

Stack The section the interpreter always wants

Time Migration Diffractions or structural dip

Depth Migration Structural dip with large lateral velocity variation

Prestack Partial Migration (PSPM) Lots of conflicting dips with different stacking velocities or large lateral velocity gradients

Depth Migration Before Stack Strong/large lateral velocity gradients that cannot be treated properly by stacking

3-D Time Migration After Stack Needed when the stack contains dipping events that are out of the profile plane (crossdips)

3-D Depth Migration After Stack Needed when the problem of strong lateral velocity gradients involves 3-D structural complexity

3-D Time Migration Before Stack When PSPM fails and stack contains crossdips

3-D Depth Migration Before Stack Everyone’s dream if computer time were available and 3-D subsurface velocity model were well known

Migration Types

Choice of Migration Techniques

Difficulty

& Expense

4 types of migration

post-stack time

post-stack depth

pre-stack time

pre-stack depth


Time migration versus depth migration

Time migration• Cannot handle large lateral velocity contrasts• Cannot handle large ray-bending effects• No need for a model

Depth migration• Can handle lateral velocity contrasts and ray-bending• Needs a velocity model

Both time and depth migration can work on post- and pre-stack data.

Time migration vs. Pre-Stack Depth Migration


Reflection on the time section C’D’ must be migrated to its true position CD.

Observations:

1. The dip angle of the reflector in the geologic section is greater than in the time section: thus migration steepens reflections.

2. The length of the reflector, as seen in the geologic section, is shorter than in the time section; thus migration shortens reflections.

3. Migration moves reflectors in the updip direction.

Exercise

From the exercise:1. Note that the dips after migration are greater than before migration.2. The deeper the event, the more migration takes place e.g at 4s the horizontal

displacement is more that 6km and the vertical displacement is 1.6seconds.3. The horizontal displacement increases with the time of event4. Displacement is a function of velocity squared (if there is a 20% error in velocity used,

event is misplaced by an error of 44%.5. Vertical displacement also increases with time.

Migration Principles


Consider the dipping reflector CD of the geologic (depth) section. We want to obtain zero offset section along the profile Ox.

The first normal incidence arrival from the dipping reflector is recorded at location A. The reflection arrival at location A is indicated by point C’.

True subsurface location depth

Zero offset seismic time section

4/)tan( 2tx tvd


}]4/)tan(1[1{ 2/122tt vtd

2/122 ]4/)tan(1/[tantan ttt v

Where:

xtt /tan


Subsurface Model

Zero offset time section

The steeper the dip the more the event moves during migration.

We see that the dipping event migrates out of the recorded section. The data on a recorded section are not necessarily confined to the subsurface below the seismic line. The converse is also true. The structure below on a seismic line may not be recorded on the seismic section.

In areas of structural dip, the line length must be chosen considering the displacements of dipping layers. The areal surface coverage must be larger than the subsurface coverage of interest.

Recording time must be long enough

What of curved reflecting surfaces? (See Diagrams)

Synclines broaden and anticlines compress

Higher velocities mean more migration and hence smaller anticlines

Why does a syncline look like a bowtie.


Migration Principles – Curved Surfaces


Before Migration After Migration


Migration unties bowties and turn them to synclines

Kirchhoff Migration

The gap in the barrier acts as Huygens’ secondary source, causing the circular wavefronts that approach the beachline. Waves recorded along the beach generated by Huygens’ secondary source (the gap in the barrier has a hyperbolic traveltime trajectory.

Kirchhoff Migration

A point that represents Huygens’ secondary source in the depth section, maps onto a diffraction hyperbola on the zero offset time section (b). The vertical axis in this section is two-way time.

Superposition of the zero offset response (b) of a discrete number of Huygens’ secondary source in the depth section (a).

Kirchhoff Migration

Superposition of the zero offset response (b) of a continuum of Huygens’ secondary source in the depth section (a).

Principles of migration based on semicircle superposition. (a) Zero-offset section (trace interval, 25m; constant velocity, 2500m/s), (b) migration, A point in time section (a) maps onto a semicircle in depth section (b).

Kirchhoff Migration

Principles of migration based on diffraction summation. (a) Zero-offset section (trace interval, 25m; constant velocity, 2500m/s), (b) migration, The amplitude at B along the flank is mapped onto apex A by the hyperbolic traveltime equation.

The first method of migration is based on the superposition of semicircles, while the second method is based on the summation of amplitudes along hyperbolic paths (diffraction summation method).

The summation is a straightforward summation of amplitudes along the hyperbolic trajectory whose curvature is governed by the velocity function.

Kirchhoff Migration

The migration scheme based on the semicircle superposition consists of mapping the amplitude at a sample in input (x,t) space of the unmigrated time section onto a semicircle in output (x,z) space. The migrated section is formed as a result of the superposition of the many semicircles.

The migration scheme based on the diffraction summation consists of searching the input data in the (x,t) space for energy that would have resulted if a diffracting source (Huygens’ secondary source) were located at a particular point in the output (x,z) space. This search is carried out bu asumming the amplitudes in the (x,t) space along the diffraction curve that corresponds to Huygens’ secondary source at each point in the (x,z) space. The result of the summation is then mapped onto the corresponding point in the (x,z) space.

Kirchhoff Migration

Velocity function:

Compute t(x). Amplitude at location B is placed on the output section at location A corresponding to the output time t(0).

Consider three factors associated with the amplitude and phase behaviour of the waveform along the diffraction hyperbola.

Amplitude at A is stronger than B (B is at an oblique angle – obliquity factor).Spherical divergence. B versus C. (wavefront at C is weaker).Restoration of resulting waveform from superposition must be restored in both phase and amplitude.

These 3 factors must be considered.

Kirchhoff Migration

The diffraction summation method of migration which incorporates these three factors, is called the Kirchhoff migration. To perform this, multiply the input data by the obliquity and spherical spreading factors. Then apply the filter with the above specifications and sum along the hyperbolic path that is defined by the velocity function equation. Place the result on the migrated section a time t(0) corresponding to the apex of the hyperbola.

The rms velocity is used.

Synthetic Migration Example

Migration (seismic): An inversion operation involvingrearrangement of seismic information elements so thatreflections and diffractions are plotted at their true locations.

Migration (seismic): An inversion operation involvingrearrangement of seismic information elements so thatreflections and diffractions are plotted at their true locations.


CMP

Time

Real-Life Example of Step-wise Migration

Migration Domain


Stack

Migration

CMP

Time

Horizontal Reflector

CMP = CDPCMP = CDP

CMP CDP

CMP CDP

Dipping Reflector

CMP = common mid point CDP = common depth point

NMO, DMO & Migration

NMO, DMO & Migration


NMO corrects for the time delay on an offset trace (assuming zero dip),DMO moves the data to the correct zero-offset trace location,Migration further moves it to the subsurface location.(after Deregowski, 1986)

Comparison PoSTM vs. PreSTM


Post Stack Time Migration Pre Stack Time Migration

Migration with 95,100 & 105% velocitiesPost stack time migration


95% velocities 100% velocities 105% velocities

3D ray traces projected on 2D display

Constant Salt Velocitiy Model

Hard Floater Model

Varying Salt Velocitiy Model

Gridded Floater Model

Inline 7496

Inline 7496

Inline 7496

Inline 7496

Ray Tracing: various salt models (identical overburden)The hard anhydrite layers cause ray-tracing problems and refractions.

Vel

oci

ty [

m/s

]

UNSEA

NSEA

CHALK

VLIE

TRIAS

SALT

ALTENA

Dep

th [

m]

PreSDM Model

3D view of an interface from a PreSDM model.

Successful Reprocessing

Existing product Reprocessing

Effect of Fold of Coverage

96 fold24 fold 48 foldCMP domainshot domain stack domain migration domain

Fold of Coverage


Seismic Data Volumes

As acquired: 300 - 700 MB per square km• Marine survey, 6 streamers @ 180 = 1080 channels;• 3000 samples per channel @ 24 bit = 9 KB / channel / shot• 10 MB per shot, 66 shots per sq. km

After processing a factor 50 - 100 data reduction: ~ 8 MB per square km

Typical survey sizes: 200 - 1000 square km• Annual acquisition volume 15000 sq km (~7500 Gb)