**Authors**

**Abstract**

Obtaining a seismic section with high temporal and spatial resolution was always one of the goals of seismic data processors and interpreters. Accurate estimation of the thicknesses of thin beds is an important tool in this regard.

The basic problem is that the wavelength of the signal must be similar in dimention to that of the bed thinness. If it is much longer than the bed thinness, the determination of interference or phase shift is less reliable. If it is much shorter, the problem is not one of a thin bed. The thin bed problem assumes that the bed is thin compared to the dominant wavelength of the wavelet.

The Rayleigh criterion states that the resolution limit of a reflection is 4, but Widess (1973) extended this limit to 8. In this research, we assume that the thinness of a thin bed is less than 8.

The differences between thin bed response and thick bed response are that thick bed response has a separate response for the top and bottom of the bed, the two wavelets do not interfere and the amplitude of the wavelet depends on reflection coefficient. But for thin beds, reflections from top and bottom of the bed interfere. The result is a signal wavelet response which approximates the time derivative of the original wavelet.

Quantitatively, bed thickness can be calculated in three ways: from the time difference of the seismic events, from the first spectral peak frequency and from the cepstral peak.

In the first method, we can calculate bed thickness from the time difference of two peaks (for two sequential traces in same polarity), or from the time differences of one peak and a trough (for two sequential traces in opposite polarity).

Widess pioneered a widely used method for quantifying thin bed thickness in 1973. Because it uses peak to trough time separation in conjunction with amplitude, this method depends on careful processing to establish the correct wavelet phase and true trace to trace amplitudes.

But by transforming the seismic data into the frequency domain via the discrete Fourier transform, the amplitude spectra delineate temporal bed thickness variability while the phase spectra indicate lateral geologic discontinuities. So, this method which is spectral decomposition uses a more robust phase independent amplitude spectrum and is designed for examining thin bed responses surveys.

Spectral decomposition unravels the seismic signal into its constituent frequencies. This allows the interpreter to see amplitude and phase tuned to specific wavelengths. Since the stratigraphy resonates at wavelengths dependent on the bedding thickness, the interpreter can image subtle thickness variations and discontinuities and predict bedding thickness quantitatively.

Thin beds produce periodic peaks and notches in the spectrum of seismic data. In classical spectral decomposition technique, the frequency of the first local maximum in the amplitude spectrum (the first spectral peak) is doubled to estimate the period of the notches which is equal to the inverse of the bed thickness.

In this study we describe a novel extension of the spectral decomposition method called cepstral decomposition.

Cepstral decomposition method can accurately measure the spacing of notches by calculating the Fourier transform of the logarithms of the spectrum. To suggest this, note that a signal with a simple echo can be represented as:

(1)

The Fourier spectral density (spectrum) of such a signal is given by:

(2)

Thus, we see from (2) that the spectral density of a signal with an echo has the form of an envelope (the spectrum of the original signal) that modulates a periodic function of frequency. By taking the logarithm of the spectrum, this product is converted to the sum of two components:

(3)

Thus, C(f) viewed as a waveform has an additive periodic component whose fundamental frequency is the echo delay In analysis of time waveforms, such periodic components show up as lines or sharp peaks in the corresponding Fourier spectrum. Therefore, the spectrum of the log spectrum would show a peak when the original time waveform contained an echo.

This new spectral representation domain was not the frequency domain, nor was it the time domain. So we call it as the “quefrency domain”, and the spectrum of the log of the spectrum of a time waveform as the “cepstrum”.

In new quefrency domain periodic notches appear as sharp peaks. The peaks are sharp and clear enough to use them for estimating thin beds thickness.

We tested the cepstral decomposition technique for estimating the thickness of thin layer on a synthetic model with different random noise levels and compared the results by that of the two conventional methods: the spectral peak method and the time difference of the seismic events.

The results indicated that cepstral decomposition method has the potential to improve the accuracy of thin bed thickness estimation from reflection seismic data.

The basic problem is that the wavelength of the signal must be similar in dimention to that of the bed thinness. If it is much longer than the bed thinness, the determination of interference or phase shift is less reliable. If it is much shorter, the problem is not one of a thin bed. The thin bed problem assumes that the bed is thin compared to the dominant wavelength of the wavelet.

The Rayleigh criterion states that the resolution limit of a reflection is 4, but Widess (1973) extended this limit to 8. In this research, we assume that the thinness of a thin bed is less than 8.

The differences between thin bed response and thick bed response are that thick bed response has a separate response for the top and bottom of the bed, the two wavelets do not interfere and the amplitude of the wavelet depends on reflection coefficient. But for thin beds, reflections from top and bottom of the bed interfere. The result is a signal wavelet response which approximates the time derivative of the original wavelet.

Quantitatively, bed thickness can be calculated in three ways: from the time difference of the seismic events, from the first spectral peak frequency and from the cepstral peak.

In the first method, we can calculate bed thickness from the time difference of two peaks (for two sequential traces in same polarity), or from the time differences of one peak and a trough (for two sequential traces in opposite polarity).

Widess pioneered a widely used method for quantifying thin bed thickness in 1973. Because it uses peak to trough time separation in conjunction with amplitude, this method depends on careful processing to establish the correct wavelet phase and true trace to trace amplitudes.

But by transforming the seismic data into the frequency domain via the discrete Fourier transform, the amplitude spectra delineate temporal bed thickness variability while the phase spectra indicate lateral geologic discontinuities. So, this method which is spectral decomposition uses a more robust phase independent amplitude spectrum and is designed for examining thin bed responses surveys.

Spectral decomposition unravels the seismic signal into its constituent frequencies. This allows the interpreter to see amplitude and phase tuned to specific wavelengths. Since the stratigraphy resonates at wavelengths dependent on the bedding thickness, the interpreter can image subtle thickness variations and discontinuities and predict bedding thickness quantitatively.

Thin beds produce periodic peaks and notches in the spectrum of seismic data. In classical spectral decomposition technique, the frequency of the first local maximum in the amplitude spectrum (the first spectral peak) is doubled to estimate the period of the notches which is equal to the inverse of the bed thickness.

In this study we describe a novel extension of the spectral decomposition method called cepstral decomposition.

Cepstral decomposition method can accurately measure the spacing of notches by calculating the Fourier transform of the logarithms of the spectrum. To suggest this, note that a signal with a simple echo can be represented as:

(1)

The Fourier spectral density (spectrum) of such a signal is given by:

(2)

Thus, we see from (2) that the spectral density of a signal with an echo has the form of an envelope (the spectrum of the original signal) that modulates a periodic function of frequency. By taking the logarithm of the spectrum, this product is converted to the sum of two components:

(3)

Thus, C(f) viewed as a waveform has an additive periodic component whose fundamental frequency is the echo delay In analysis of time waveforms, such periodic components show up as lines or sharp peaks in the corresponding Fourier spectrum. Therefore, the spectrum of the log spectrum would show a peak when the original time waveform contained an echo.

This new spectral representation domain was not the frequency domain, nor was it the time domain. So we call it as the “quefrency domain”, and the spectrum of the log of the spectrum of a time waveform as the “cepstrum”.

In new quefrency domain periodic notches appear as sharp peaks. The peaks are sharp and clear enough to use them for estimating thin beds thickness.

We tested the cepstral decomposition technique for estimating the thickness of thin layer on a synthetic model with different random noise levels and compared the results by that of the two conventional methods: the spectral peak method and the time difference of the seismic events.

The results indicated that cepstral decomposition method has the potential to improve the accuracy of thin bed thickness estimation from reflection seismic data.

**Keywords**