Frequency and Bandwidth: their relationship to Seismic Resolution

Tom Wilson, Department of Geology and Geography

Environmental and Exploration Geophysics II

[email protected]

Department of Geology and GeographyWest Virginia University

Morgantown, WV

Frequency and Bandwidth: Frequency and Bandwidth: their relationship to their relationship to Seismic ResolutionSeismic Resolution


How can we build this wavelet?


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

Composite Wavelet(sum of cosines)

-0.4-0.2

00.20.40.60.8

11.2

-0.06 -0.04 -0.02 0 0.02 0.04 0.06Time (seconds)

Ampl

itude


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

wavelet.

O. Ilmaz, 1987


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.


Time domain

Frequency domain

O. Ilmaz, 1987


Time-domain wavelets Zero Phase Minimum

Phase

Individual frequency components

Amplitude spectrum

Phase spectrum

Hilterman, 1985


Extracting information about wavelet frequency content from an isolated reflection event.The dominant period (c) 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 (b) 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


Let’s come back to this issue in a minute, but first let’s pull some ideas together to develop a basic understanding of how the seismic signal arises in terms of reflection coefficients and wavelets.

Exxon in-house course

notes

Shape of up-going wave is

reversed

In space the leading positive cycle changes to a leading negative cycle



notes

Shape of up-going wave is

unchanged

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

notes

negative

positive


As the two layers move closer and closer together we get to a point where the second cycle in the wavelet reflected from the top of the layer overlaps with the arrival of 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.


notes


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) attains peak amplitude. Exxon in-

house course notes


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


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

Side lobe

trough

peak

Reflection Coefficients


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/c. However, the amplitude of the composite continues to drop. But not the period.


notes


The peak-to-trough time equals c/2.

Side lobe

trough

peak

Seismic Wavelet

Maximum Constructive Interference

Two-way interval time separating

reflection coefficients is

c/2


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

similar to that which you will generate in lab today.


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


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 (c/2). If the interval velocity of the layer in question is known, the dominant period can be converted into the tuning thickness.


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


c/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.


What is the amplitude spectrum of wavelet #5?

Ilmaz, 1987Broader spectra produce sharper, shorter duration wavelets


Spectral bandwidth, wavelet duration in the time domain and resolution. C is only one parameter that affects resolution. b is also an important parameter.

Hilterman, 1985

Greatest Bandwidth

Smallest Bandwidth


Physical nature of the seismic responseHilterman, 1985

The Convolutional Model


The output is a superposition of reflections from all acoustic interfaces


notes

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

notes



Subsurface structure - North Sea

One additional topic to consider in general is wavelet deconvolution and how 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 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. Deconvolution improves seismic resolution and simplifies interpretation.


North Sea Seismic display after deconvolution. The geometrical

interrelationships between reflectors are clearly portrayed.

Neidel, 1991


Consider the following problem -You are given the seismic wavelet shown

below.

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?


The minimum phase wavelet has its energy concentrated toward the front end of the wavelet. The amplitude of the disturbance decays exponentially. This wavelet is a causal wavelet and the location of the reflection coefficient is placed at the wavelet onset, which can be difficult for the interpreter to pick.


The zero phase wavelet is symmetrical. This wavelet is centered over the reflection coefficient. The zero phase wavelet is produced through data processing and is not generated naturally. It is non causal - half of the wavelet arrives before the reflector appears in time. It is easy for an interpreter to pick reflection times using the zero phase wavelet since highest amplitude occurs at the reflection boundary.


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


The default wavelet in Struct is the Ricker wavelet. The Ricker wavelet is zero phase.

Hilterman, 1985


If you haven’t already, finish reading chapter 4.

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Frequency and Bandwidth: their relationship to Seismic Resolution