COMPUTING INSTANTANEOUS ATTRIBUTES PROGRAM instantaneous_attributes Computation flow chart The input to program instantaneous_attributes is a time-domain seismic amplitude volume. The input could be either the original seismic amplitude, a structure-oriented filtered amplitude from program sof3d, spectrally-balanced amplitude from program spec_cmp or spec_cwt, or even the amplitude processed by a commercial software package. The output files include various types of instantaneous attributes, which are Hilbert transformed data, instantaneous envelope, instantaneous phase, instantaneous frequency, wavelet phase, wavelet frequency, amplitude volume technique (AVT) data, sweetness, and unwrapping phase. Attribute-Assisted Seismic Processing and Interpretation Page 1
Computing instantaneous attributes The program instantaneous_attributes is launched from aaspi_util GUI under Single Trace Attributes menu or by typing aaspi_instantaneous_attributes in the command line: Attribute-Assisted Seismic Processing and Interpretation Page 2
Clicking instantaneous_attributes generates the following GUI: Attribute-Assisted Seismic Processing and Interpretation Page 3
Computing instantaneous attributes Use the browser on the first line to choose the input seismic data file (channel.h). (2) is the output of its Hilbert transform. (3) is the output of instantaneous envelope. (4) is the output of instantaneous phase. (5) is the output of instantaneous frequency. (6) is the output of the wavelet phase. (7) is the output of the wavelet frequency. (8) is the output of amplitude volume technique (AVT). (9) is the output of the sweetness. (10) is the output of the unwrapping phase. Attribute-Assisted Seismic Processing and Interpretation Page 4
Parameters for Instantaneous Attributes Calculation The image below shows the parameters tab used for instantaneous attributes calculation. (1) is AVT window size, which defines the time samples used for AVT (amplitude volume technique) calculation. The AVT result has lower frequency with bigger window size. (2) is the median filter window size. The result is more smoothed with bigger window size. (3) is the window size for wavelet-based attributes calculation. (4) is the minimum time for instantaneous attributes calculation. (5) is maximum time for instantaneous attributes calculation. Attribute-Assisted Seismic Processing and Interpretation Page 5
Input seismic amplitude The seismic amplitude of GSB survey is used as example. A seismic line (inline=3141) is shown is the following. Hilbert transform The result of Hilbert transform is shown in the following. Attribute-Assisted Seismic Processing and Interpretation Page 6
We also illustrate theory of Hilbert transform in the following. Theory of Hilbert transform In general, any continuous function, u(t), can be thought of as the real part of an analytic function, A(t), where A(t)=u(t)+ju H (t), where j=(-1) 1/2 is the imaginary unit and u H (t) is called the Hilbert transform. The Hilbert transform can be computed in either the time domain or the frequency domain. In the time domain, the Hilbert transform is defined as H 1 u ( t) ut (2n 1) t 2n 1 n, where the operator is a sum over the reciprocal of odd integers. Note that the limits of the operator can be quite large, requiring it to be truncated for computational efficiency. An alternative way to compute the Hilbert transform is to compute, U(ω), the Fourier transform of u(t) U ( ) F u( t ) and then rotate the complex Fourier components by 90 0, i.e., by multiplying by the imaginary unit, i=exp(iπ/2) H u ( t) F iu ( ). 1 Attribute-Assisted Seismic Processing and Interpretation Page 7
Instantaneous envelope, phase, and frequency With the input seismic amplitude and the Hilbert transform result, the user can calculate the instantaneous envelope, instantaneous phase, and instantaneous frequency. The theory is shown in the following. Theory of instantaneous envelope, phase, and frequency The instantaneous envelope, e(t), is calculated with 2 H 2 1/2 e( t) u( t) u ( t) The instantaneous phase, φ(t), is calculated with ( t) ATAN2 u H ( t), u( t) In principal, the instantaneous frequency is the derivative of the instantaneous phase () ( ) 2 d f t t dt. Unfortunately, this formula is useless since there are discontinuities in φ(t) as it cycles from +π to -π. To get around this difficulty, Taner et al. (1979) use the definition of φ(t) and use the chain rule to obtain H H du ( t) H du( t) du ( t) H du( t) H datan2 u ( t), u( t) u( t) u ( t) u( t) u ( t) f( t) 2 2 dt dt 2 dt dt. 2 H 2 2 dt u( t) u ( t) e () t The above equation requires computation of derivatives, which are most easily computed using the Fourier transform. If the forward Fourier transform is U( ) F u( t) and its inverse u( t) F U( ) Then du() t F dt and 1 1 iu ( ) H du () t 1 H ( ) 1 F iu F iiu ( ) F 1 U ( ) dt Attribute-Assisted Seismic Processing and Interpretation Page 8
The results of instantaneous envelope of the GSB input seismic line are shown in the following, and instantaneous phase, and instantaneous frequency. Attribute-Assisted Seismic Processing and Interpretation Page 9
Wavelet attributes The user can calculate the wavelet phase and wavelet frequency with the input seismic amplitude. Bodine first introduced the response attributes. He argued that since most of the signal energy in a trace is found in the vicinity of the envelope peaks, the reflection events phase and frequency could be more accurately described by assigning them to the value at envelope peaks. Here we use a more descriptive term of wavelet attributes by Taner. The workflow is: 1) Search the local envelope maxima, 2) Search the local envelope minima, 3) Assign the phase and frequency at the maximum to all values between the adjacent minima on either side. The following figures are the wavelet phase and wavelet frequency of the inline data. Attribute-Assisted Seismic Processing and Interpretation Page 10
Sweetness Sweetness is an attribute that was developed for identifying sands and sandstones. It is derived by dividing reflection strength by the square root of instantaneous frequency. In the program instantaneous_attributes, we use the instantaneous envelope as the reflection strength. The sweetness of inline 3141 is shown in the following. Amplitude Volume Technique (AVT) The AVT attribute produced by the program instantaneous_attributes helps to identify the different geologic features, such as fault, channel, carbonate, reflector unconformities and terminations. It is calculated by the root square of the average of the square of envelopes that are Attribute-Assisted Seismic Processing and Interpretation Page 11
within a defined analysis window, followed by the Hilbert transform. The AVT result of inline 3141 is shown in the following. Unwrapping phase In the program instantaneous_attributes, we followed the method of Aldo Vesnaver (2017) to calculate the unwrapping phase, which avoids the discontinuities. The theory is listed in the following. Attribute-Assisted Seismic Processing and Interpretation Page 12
Theory of unwrapping phase The complex trace u(t) could be expressed by instantaneous envelope e(t) and instantaneous phase φ(t), u( t) e( t)exp i ( t ) A normalized complex trace is defined by dividing the complex trace by its envelope e(t): n( t) ct () exp i ( t) et () Then we can get 2 * n( t) n( t) n ( t )=1 The derivative of the normalized complex trace is n ' t i t i ' t in t ' t ( ) exp ( ) ( ( )) ( ) ( ) By multiplying * - in ( t) on both sides of the above equation, * ' 2 * ' ' in ( t) n ( t) i n ( t) ( t) ( t ) So the unwrapping instantaneous phase could be obtained as an ' ( t)= t ( ) 0 d The unwrapping phase of inline 3141 is shown in the following. Attribute-Assisted Seismic Processing and Interpretation Page 13
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