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1
Tom Wilson, Department of Geology and Geography
Environmental and Exploration Geophysics II
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
Tom Wilson, Department of Geology and Geography
2
Tom Wilson, Department of Geology and Geography
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
Tom Wilson, Department of Geology and Geography
The examples below illustrate the effect of increasing the frequency range or bandwidth of the wavelet.
O. Ilmaz, 1987See sumofcosines.xls
3
Tom Wilson, Department of Geology and Geography
The following simple example helps illustrate the concept of an amplitude spectrum. Below is
a signal consisting of two sinusoids.
Tom Wilson, Department of Geology and Geography
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.
4
Tom Wilson, Department of Geology and Geography
Time domain
Frequency domain
O. Ilmaz, 1987
Tom Wilson, Department of Geology and Geography
Time-domain waveletsZero Phase Minimum Phase
Individual frequency components
Amplitude spectrum
Phase spectrum
Hilterman, 1985
5
Tom Wilson, Department of Geology and Geography
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
Tom Wilson, Department of Geology and Geography
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
6
Tom Wilson, Department of Geology and Geography
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
Tom Wilson, Department of Geology and Geography
Exxon in-house course notes
Shape of up-going wave is unchanged
Positive reflection coefficient
These are minimum phase wavelets
7
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
Tom Wilson, Department of Geology and Geography
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).
Exxon in-house course notes
Decrease the two-way travel time between reflection coefficients <Lead cycle
<Follow cycleLead cycle >
Reflection from the base of the layer
8
Tom Wilson, Department of Geology and Geography
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.
Exxon in-house course notes
Sum of reflection amplitudes from overlap in the top
and base reflections
Tom Wilson, Department of Geology and Geography
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
9
Tom Wilson, Department of Geology and Geography
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
Tom Wilson, Department of Geology and Geography
10
Tom Wilson, Department of Geology and Geography
Environmental and Exploration Geophysics II
Department of Geology and GeographyWest Virginia University
Morgantown, WV
The The ConvolutionalConvolutional Model and Seismic Model and Seismic Resolution (continued)Resolution (continued)
Tom Wilson, Department of Geology and Geography
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.
Exxon in-house course notes
11
Tom Wilson, Department of Geology and Geography
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
Tom Wilson, Department of Geology and Geography
Model of a thinning layer
Low velocity sand
15,000 fps
11,300 fps
19,000 fps
12
Tom Wilson, Department of Geology and Geography
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
Tom Wilson, Department of Geology and Geography
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
13
Tom Wilson, Department of Geology and Geography
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
Tom Wilson, Department of Geology and Geography
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.
14
Tom Wilson, Department of Geology and Geography
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
Tom Wilson, Department of Geology and Geography
What is the amplitude spectrum of wavelet #5?
Ilmaz, 1987
Broader spectra produce sharper, shorter duration wavelets
15
Tom Wilson, Department of Geology and Geography
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
Tom Wilson, Department of Geology and Geography
Physical nature of the seismic responseHilterman, 1985
The Convolutional Model ( ) ( ) ( )s t r w t dτ τ τ∞
−∞= −∫
16
Tom Wilson, Department of Geology and Geography
The output is a superposition of reflections from all acoustic interfaces
Exxon in-house course notes
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
Tom Wilson, Department of Geology and Geography
The wavelet in this case is also minimum phase
17
Tom Wilson, Department of Geology and Geography
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
Tom Wilson, Department of Geology and Geography
Potential hydrocarbon trap?
Below is the synthetic seismic response computed for the North Sea model.
Neidel, 1991
Consider part 2 of the handout
18
Tom Wilson, Department of Geology and Geography
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
Tom Wilson, Department of Geology and Geography
Deconvolution is a filter operation which compresses and simplifies the shape of the seismic wavelet. Deconvolutionimproves seismic resolution and simplifies interpretation.
19
Tom Wilson, Department of Geology and Geography
North Sea Seismic display after deconvolution. The geometrical interrelationships between
reflectors are clearly portrayed.
Neidel, 1991
Tom Wilson, Department of Geology and Geography
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?
20
Tom Wilson, Department of Geology and Geography
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
Tom Wilson, Department of Geology and Geography
Zero versus minimum
Hilterman, 1985
21
Tom Wilson, Department of Geology and Geography
If you haven’t already … finish reading chapter 4!