amplitude, frequency and bandwidth and their relationship...

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Tom Wilson, Department of Geology and Geography

Environmental and Exploration Geophysics II

[email protected]

Department of Geology and GeographyWest Virginia University

Morgantown, WV

Amplitude, Frequency and Amplitude, Frequency and Bandwidth and their relationship to Bandwidth and their relationship to

Seismic ResolutionSeismic Resolution


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The range of frequencies present in the wavelet controls its ability to resolve the top and bottom of a layer of given thickness.

Recall our general introduction to the concept of the wavelet earlier in the semester.

The wavelet or transient mechanical disturbance generated by the source can be thought of as a superposition or summation of sinusoids with varying frequency and amplitude.

Hilterman, 1985


The examples below illustrate the effect of increasing the frequency range or bandwidth of the wavelet.

O. Ilmaz, 1987See sumofcosines.xls

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The following simple example helps illustrate the concept of an amplitude spectrum. Below is

a signal consisting of two sinusoids.


Each sinusoid is associated with a specific frequency. There are two frequency

components. The 32 sample per cycle component has a frequency of 4 and the 8 samples per cycle component has a frequency of 16. The

amplitude of the 32 sample/cycle component is twice that of the 8 sample/cycle component.

The frequency spectrum (above) of the “signal” at the top of the previous slide is an equivalent representation of the signal.

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Time domain

Frequency domain

O. Ilmaz, 1987


Time-domain waveletsZero Phase Minimum Phase

Individual frequency components

Amplitude spectrum

Phase spectrum

Hilterman, 1985

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Extracting information about wavelet frequency content from an isolated reflection event.

The dominant period (τc) of the response corresponds to the time from one peak to the next or from one trough to the next. The reciprocal of this dominant period is a measure of the dominant frequency (fc) of the signal or wavelet spectrum.

The reciprocal of the half-width of the response-envelop (τb) provides an estimate of the bandwidth (fb) of the signal spectrum.

Hilterman, 1985


The dominant frequency and bandwidth measured from the time-domain representation of the signal wavelet can be used to provide a sketch of the wavelet spectrum.

Just as importantly these measures can be related directly to the resolution properties of the seismic wavelet.

Hilterman, 1985

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Review your basic understanding of how the composite seismic signal arises in terms of horizon reflection coefficients and the seismic

wavelet. The view below provides a temporal view of reflection shape.

Exxon in-house course notes

Shape of up-going wave is

reversed

Negative reflection coefficient



Shape of up-going wave is unchanged

Positive reflection coefficient

These are minimum phase wavelets

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Tom Wilson, Department of Geology and Geography Exxon in-house course notes

negative

positive

Positive reflection coefficient

Negative reflection coefficient

<Lead cycle

<Follow cycle

<Lead cycle

<Follow cycle


If the two layers are located closer together we get to a point where the second cycle in the reflected wavelet from the top of the layer overlaps the lead cycle in the wavelet reflected from the base of the layer. This occurs at two-way time equal to 1/2 the dominant period of the wavelet (or ½ the dominant cycle).


Decrease the two-way travel time between reflection coefficients <Lead cycle

<Follow cycleLead cycle >

Reflection from the base of the layer

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At this point there is maximum constructive interference between the reflections from the top and bottom of the layer. The composite reflection event (at

right above) reaches maximum negative value in this case.


Sum of reflection amplitudes from overlap in the top

and base reflections


The peak period of the wavelet can be determined using peak-to-trough times which correspond to one half the dominant period of the wavelet. Multiply those times by two to get the dominant period.

Dominant (or peak) frequency and wavelet phase (shape).

Referred to as 0-phase since all

frequency components are in phase

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Maximum constructive interference illustrated for the zero phase wavelet. The peak-to-trough time equals τc/2, which also equals delay time between consecutive reflection events

Side lobe

trough

peak

Reflection Coefficients


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Environmental and Exploration Geophysics II

[email protected]

Department of Geology and GeographyWest Virginia University

Morgantown, WV

The The ConvolutionalConvolutional Model and Seismic Model and Seismic Resolution (continued)Resolution (continued)


Once the separation in time drops to less than half the dominant period of the wavelet destructive interference in the reflections from the top and bottom of the layer will occur.

However, as the layer continues to thin, the dominant period of the composite reflection event does not drop below 1/τc. The amplitude of the composite continues to drop. But not the period.


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The peak-to-trough time equals τc/2.

Side lobe

trough

peak

Seismic Wavelet

Maximum Constructive Interference

Two-way interval time separating

reflection coefficients is τc/2


Model of a thinning layer

Low velocity sand

15,000 fps

11,300 fps

19,000 fps

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These amplitude relationships are summarized below in the model seismic response of a thinning layer similar to that

shown in the preceding slides.

Zero phase wavelet


The amplitude difference -trough-to-peak remains constant for two-way travel times much greater than half the dominant period.

As the top and bottom of the layers merge closer and closer together, the lead cycle in the reflection from the base of the layer overlaps with the follow-cycle in the reflection from the top and the amplitude of the composite reflection event begins to increase.

Thickness =Vt/2

Dest

ruct

ive

inte

rfere

nce

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Layer thickness is simply Vt/2, where t is the two-way interval transit time. Tuning occurs at two-way times equal to one-half the dominant period (τc/2). If the interval velocity of the layer in question is known, the dominant period can be converted into the tuning thickness.

In this plot the conversion to thickness has already been made. Compute τc.

Let layer thickness = d; then

d=? De

stru

ctiv

e in

terfe

renc

e


Difference of arrival time between the reflections from the top and bottom of the layer decreases abruptly at about 8 milliseconds.

8 milliseconds represents the two-way travel time through the layer; it is also the time at which tuning occurs and is half the dominant period of the seismic wavelet.

8 milliseconds is τc/2 and the two way time through the layer. Thus, τc/4 is the one-way time through the layer.

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τc/4, the one-way time through the layer, equals 4 milliseconds. The interval velocity in the layer is 11,300 f/s. Hence, the thickness of the layer at this point is ~45 feet.

This is the tuning thickness or minimum resolvable thickness of the layer obtainable with the given seismic wavelet.

11,300 f/s * 0.004s = 45.2 feet


What is the amplitude spectrum of wavelet #5?

Ilmaz, 1987

Broader spectra produce sharper, shorter duration wavelets

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Spectral bandwidth, wavelet duration in the time domain and resolution. τC is only one parameter that affects

resolution. τb is also an important parameter.

Hilterman, 1985

Greatest Bandwidth

Smallest Bandwidth


Physical nature of the seismic responseHilterman, 1985

The Convolutional Model ( ) ( ) ( )s t r w t dτ τ τ∞

−∞= −∫

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The output is a superposition of reflections from all acoustic interfaces


The seismic response is dominated by reflections from layers 1 and 2. We see two prominent events. They are delayed because the wavelet phase is minimum.

1

2a

2b2a

1

2b


The wavelet in this case is also minimum phase

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Subsurface structure - North Sea

One additional topic to consider is the process of wavelet deconvolution. As you’ve seen already, wavelet shape can affect

geologic interpretations …. Consider the following structural model

Neidel, 1991


Potential hydrocarbon trap?

Below is the synthetic seismic response computed for the North Sea model.

Neidel, 1991

Consider part 2 of the handout

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Consider the effect of wavelet shape on the geologic interpretation of seismic response. In the case shown below, the primary reflection from the base of the Jurassic shale crosses a side-lobe in the wavelet reflected from the overlying basal Cretaceous interval.

Neidel, 1991


Deconvolution is a filter operation which compresses and simplifies the shape of the seismic wavelet. Deconvolutionimproves seismic resolution and simplifies interpretation.

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North Sea Seismic display after deconvolution. The geometrical interrelationships between

reflectors are clearly portrayed.

Neidel, 1991


Any questions about today’s exercises?

Using the estimation procedure discussed in class today measure the appropriate feature on the above seismic wavelet and answer the following questions:

What is the minimum resolvable thickness of a layer having an interval velocity of 10,000fps? Show work on your handout

What is the phase of the wavelet? Why do you say that?

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The zero-phase wavelet is also considered to have higher resolving power. It is generally more compact than the equivalent minimum phase wavelet and is, overall, easier to interpret.

The exploration data is in a zero phase format.

Hilterman, 1985


Zero versus minimum

Hilterman, 1985

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If you haven’t already … finish reading chapter 4!

amplitude, frequency and bandwidth and their relationship...

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