Arab J Geosci92: )6102( DOI 10.1007/s12517-015-2044-8
SHORT COMMUNICATION
Review of instantaneous, wavelet, and weighted seismic attributes along with a computational library Khalid Amin Khan 1 & Gulraiz Akhter 2
Received: 6 January 2015 / Accepted: 10 September 2015 # Saudi Society for Geosciences 2015
Abstract Complex trace analysis is the most widely used method for transforming post-stack seismic traces into a number of instantaneous seismic attributes which are used to make a variety of geological interpretations. Several useful interpretations require a combination of these attributes to be computed and displayed simultaneously which demand high performance and large memory resources. This paper reviews instantaneous, wavelet, and weighted average attributes and presents a digital signal processing library for their efficient computation by optimizing performance and memory. Keywords Hilbert transform . Seismic attributes . Interpretation . Software library
Introduction The seismic method was developed in the 1930s as a major tool to delineate the geometry of subsurface geological structures associated with hydrocarbons. In the late 1960s, the advent of digital technology provided better resolution and dynamic range, enabling monitoring of slightest changes in amplitude. The implementation of digital signal processing algorithms further revolutionized this method, allowing more geologic information to be extracted from seismic traces. In the 1970s, complex trace analysis was introduced to seismic
* Khalid Amin Khan
[email protected] 1
K-tron Research Inc., 347-A Westridge I, Rawalpindi, Pakistan
2
Department of Earth Sciences, Quaid-i-Azam University, Islamabad, Pakistan
(Taner et al. 1979). which laid the foundations of post-stack seismic attributes. Since then, there have been tremendous advancements in attribute analysis. Chopra and Marfurt (2005) have given a complete overview of the historic developments related to seismic attributes. A number of new attributes have been developed and classified in a number of ways, such as pre-stack and post-stack attributes; instantaneous, window, and trace-to-trace attributes; and time, amplitude, frequency, and attenuation-derived attributes (Brown 1996; Brown 2001; Taner 2001). The basic information of seismic trace is time, amplitude, frequency, and attenuation. An attribute is basically a derivative of one or more basic information of the trace. Attributes quantify specific data characteristics and, therefore, represent subsets of the total information. They describe or highlight specific information from seismic data. Attribute computations decompose seismic data into constituent attributes. There are no rules governing computation of attributes. They can be derived from pre-stack as well as post-stack seismic data. Thus, any quantity calculated from seismic data can be considered as an attribute. Although direct relationships have not been established between all of the attributes and the physical and geological characteristics of the Earth, the geological significance of a number of attributes has been identified. Today, seismic attributes are considered as a reliable tool for prediction of lithology, porosity, fluid content (Chopra and Marfurt 2008a). and reservoir characterization (Kalkomey 1997). They are successfully used by geologists, geophysicists, and petrophysicists to map features from basin to reservoir scale (Chopra and Marfurt 2008b). This paper presents an efficient signal processing library for computation of post-stack instantaneous and wavelet attributes from complex trace analysis. It can be implemented is seismic interpretation applications for quick visualization of multiple attributes.
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Hilbert transform and complex trace analysis Hilbert transform is commonly used in electrical engineering, signal processing (Gabor 1946; Bracewell 1965; Oppenheim and Schafer 1975). and seismic analysis (Famback 1975; Taner and Sherif 1977). It is generally introduced as a convolution between a real signal f(t) and 1/πt as given below: 1 1 hðt Þ ¼ *f ðt Þ ¼ P πt π
Z∞ −∞
f ðτ Þ dτ t −τ
ð1Þ
where P is the Cauchy principal value and h(t) is the Hilbert transformed −90° phase rotated imaginary part of the real signal. In seismic method, the Hilbert transform is considered as a starting point for the complex trace analysis, which serves as the basis for computation of post-stack instantaneous seismic attributes (Taner et al. 1979). The recorded seismic trace is the real part of the complex analytical signal and represents the kinetic component of seismic energy. The computed imaginary or conjugate part, called the quadrature trace, represents the potential component of seismic energy. These real f(t) and imaginary h(t) traces can be expressed in terms of a timedependent amplitude A(t) and phase θ(t) as f ðt Þ ¼ Aðt Þcosθðt Þ
ð2Þ
hðt Þ ¼ Aðt Þsinθðt Þ
ð3Þ
In digital signal processing, the Hilbert transform is applied to discrete data (Kak 1970; Cizek 1970). Some variants of this transform are used in geophysics (Spaendonck et al. 2002; Luo et al. 2003). It is usually applied in the form of a windowed Hilbert transform having a finite length in time. This simply involves the convolution of the input real trace with a windowed version of the ideal Hilbert transformer to get the imaginary trace. A normalized windowed Hilbert transformer or wavelet operator is shown in Fig. 2.
Instantaneous attributes The instantaneous attributes are computed sample by sample from the complex seismic trace (White 1991). Robertson and Nogami (1984) have discussed the interpretive significance of the basic instantaneous attributes. In this section, we will focus on the computation and geological significance of all major instantaneous attributes. These attributes are derived one by one from the complex trace analysis. In the next sections, we will discuss wavelet attributes and weighted average attributes, derived from these instantaneous attributes. Reflection strength (instantaneous amplitude)
F ðt Þ ¼ f ðt Þ þ ihðt Þ
ð4Þ
F ðt Þ ¼ Aðt Þcosθðt Þ þ iAðt Þsinθðt Þ ¼ Aðt ÞeiθðtÞ
ð5Þ
Reflection strength, also called the trace envelope, represents the total instantaneous energy of the complex trace independent of the phase and is computed as the modulus of the complex trace as given below: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Aðt Þ ¼ f 2 ðt Þ þ h2 ðt Þ ¼ j F ðt Þj ð6Þ
The relationship between real, imaginary, and complex trace is graphically illustrated in Fig. 1.
It represents acoustic impedance contrast and is a good indicator of bright spots, gas accumulation, sequence
Now, the complex or analytical trace F(t) is mathematically expressed as
Fig. 1 Complex trace analysis (modified from Taner et al. 1979)
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conditions. It is mathematically given by (Cohen 1989; Cohen and Lee 1990); 1 d σð t Þ ¼ ð10Þ lnðAðt ÞÞ 2π dt It represents seismic data bandwidth sample by sample and is one of the high-resolution character correlators. It shows overall effects of absorption and seismic character changes. Fig. 2 Normalized Hilbert transformer wavelet
Instantaneous quality factor boundaries, depositional environments, thin-bed tuning effects, unconformities, faults, and major changes of lithology. Instantaneous phase The phase information is independent of trace amplitudes and relates to the propagation phase of the seismic wave front. It is mathematically expressed as θðt Þ ¼ tan−1 ½hðt Þ= f ðt Þ
Instantaneous frequency Instantaneous frequency is the rate of change of phase over time. Derivative of instantaneous phase is instantaneous angular frequency given by ð8Þ
Thus, the instantaneous frequency is given by υðt Þ ¼ ωðt Þ=2π
qðt Þ ¼ υðt Þ=2σðt Þ
ð11Þ
ð7Þ
As wave fronts are defined as lines of constant phase, the phase attribute is a physical attribute and can be effectively used as a discriminator for geometrical shape classifications. Thus, it is the best indicator of lateral continuity.
ωðt Þ ¼ dθðt Þ=dt
Instantaneous quality factor is a transmissive attribute, similar to the interval and instantaneous velocities. It is also a physical attribute with a strong relation to porosity, permeability, and fracture and indicates relative absorption characteristics of beds. It is given by
ð9Þ
Instantaneous frequency is a physical attribute, influenced by the bed thickness, and can be used for seismic character correlation. Zeng (2010) has discussed in detail the geologic significance of instantaneous frequency. It is a good indicator of hydrocarbons by low-frequency anomaly, fracture zone by low-frequency zones, chaotic reflection zone, edges of low impedance thin beds, sand/shale ratio, and bed thickness with higher frequencies indicating sharp interfaces or thin shale bedding and lower frequencies indicating sand-rich bedding. Instantaneous bandwidth Instantaneous bandwidth is a statistical measure of the seismic wavelet and relates to various physical
Dominant frequency Root mean square (RMS) frequency of the amplitude spectrum represents a biased mean towards the dominant frequency band. It is similar to the instantaneous frequency, except that it corresponds to the RMS frequency of the amplitude spectrum or centroid of the power spectrum of the seismic wavelet. Barnes (1992) has discussed the characteristics of dominant frequency together with instantaneous frequency and bandwidth. It is well suited for the investigation of timevarying spectral changes and can be used along with instantaneous frequency to recognize a low-frequency shadow, lending more confidence to an interpretation. Dominant frequency is mathematically expressed as (Cohen 1989); pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f d ðt Þ ¼ υ 2 ðt Þ þ σ2 ðt Þ ð12Þ
Amplitude acceleration It is also called envelope second derivative and gives a measure of sharpness of the envelope peak, which may be more useful as a principal attribute display. It is given by Ac ðt Þ ¼
d lnðAðt ÞÞ dt 2
ð13Þ
It indicates effects of absorption and sharp changes of lithology (Taner et al. 1994) and provides a good presentation of subsurface image by showing all reflecting interfaces visible within seismic bandwidth.
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Cosine of phase A cosine of phase display is similar to seismic data processed with a short-gate automatic gain control (White 1991). It has the same uses as instantaneous phase, but has the advantage of smoothly oscillating between positive and negative values instead of discontinuous wrapping. From (2), the cosine of the instantaneous phase is given by cosθðt Þ ¼ f ðt Þ=Aðt Þ
ð14Þ
Wavelet attributes The wavelet attributes are similar to instantaneous attributes, but their values are computed at the peaks of the instantaneous wavelet envelope. The physical meaning of all the wavelet attributes is essentially the same as their instantaneous counterparts. These attributes are computed at the local maximum or peak of the amplitude envelope; this value is held constant and applied to all samples between minima in the amplitude envelope trace (Bodine 1986). This process is repeated for each envelope peak; as a result, these attributes have a blocky appearance. Some instantaneous attributes are difficult to interpret due to their spiky appearance; computing their blocky wavelet equivalents makes them easy to interpret. Apparent polarity Apparent polarity is defined as the sign of the recorded trace where the reflection strength trace has a local maximum. This value is applied to all time samples between minima. It is mathematically given by APn1→n2 ¼ Aðt Þm *Sign f ðt Þm ð15Þ where m is the local maximum between two minima n1 to n2 in the amplitude envelope. Apparent polarity is extremely sensitive to data quality. It can be helpful in distinguishing between different types of amplitude anomalies such as bright spots showing negative polarity for reservoir top reflections due to lower acoustic impedance of gas-accumulated clastic sediments than the surrounding beds and flat spots showing positive polarity for reflections from gas-oil or gas-water interfaces. Response phase and response frequency Response phase is defined as the instantaneous phase calculated at the peak of the amplitude envelope and applied to all samples between minima. It is mathematically given by θrn1→n2 ¼ θðt Þm
ð16Þ
where m is local maximum between two minima n1 to n2 in the amplitude envelope. Similarly, response frequency is computed from instantaneous frequency and given by υrn1→n2 ¼ υðt Þm
ð17Þ
Response phase and response frequency attempt to extract physically meaningful phase and frequency information about the localized seismic wavelet. They may be used to distinguish between pay zones and non-pay zones with a similar amplitude response (Bodine 1986). A response phase display will emphasize the dominant phase characteristics of the reflectors and is also useful for detecting phase changes associated with lateral fluid content or lithologic changes. Similarly, response frequency will emphasize the dominant frequency characteristics of the reflectors and may be useful in detecting the effects of frequency absorption related to fluid content, fracturing, or changing depositional environments.
Weighted average attributes Instantaneous attributes have excellent resolution but are prone to noise and spikes, making them difficult to interpret. The weighted average attributes remove spikes and reduce rapid and confusing variations. This improves interpretability at the expense of resolution (Barnes 2000). These attributes are computed by averaging in a small window weighted by instantaneous amplitude or instantaneous power which is amplitude squared. They produce a local measure that equals a Fourier spectral average. Weighted average attributes are commonly computed for instantaneous frequency and instantaneous bandwidth. The weighted averaging is accomplished with an 11-point Hamming window. The resulting attributes are smooth, spike-free in appearance and highlight the continuity of a signature on a seismic section.
Attributes computation library Post-stack seismic data exists in the form of a 2D section or a 3D volume comprising of thousands of traces. The attributes need to be computed for all traces in a large seismic dataset. This demands system resources in the form of memory and computation time. Meaningful geologic interpretation requires a combination of attributes to be displayed simultaneously or efficient switching between the displays of various attributes. In the previous sections, it can be seen that attributes are computed successively from one another. This will require a lot of computation time to interactively switch between displays of
Arab J Geosci92: )6102( Fig. 3 Block diagram of seismic attributes computation library
Page 5 of 7 29 Input: Seismic Trace (Real)
Convolution
Hilbert Transformer Wavelet
Quadrature Trace (Imaginary) Reflection Strength
Output Memory Buffers Fixed
Instantaneous Phase
Floating
Instantaneous Frequency Instantaneous Bandwidth Inst. Quality Factor Dominant Frequency Amplitude Acceleration Cosine of Phase
Apparent Polarity Response Phase Response Frequency Weighted Average
Outputs: Computed Attributes
attributes. On the other hand efficient attribute displays can be generated by initial computation of all required attributes and storing them in separate memory arrays. This approach will require a lot of memory resources. To resolve the above mentioned problems, an efficient signal processing library for computation of attributes has been developed, by compromising between computation time and memory resources. Figure 3 shows the block diagram of this library. It can be seen that the imaginary quadrature trace is computed from the input seismic trace by convolving it with Fig. 4 A seismic trace, its computed quadrature trace, and instantaneous as well as wavelet attributes
the Hilbert transformer wavelet. The diagram also shows a cascaded array of modules for successive computation of various attributes from one another. Thus, some attributes are derived from one or more attributes. Finally, all computed attributes are available as outputs. For a large seismic dataset, computation and display of derived attributes will require a longer computation time. Thus, interactively switching between displays of derived attributes will not be feasible unless these attributes are initially computed and stored in memory buffers. In this way, the
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Fig. 5 Seismic section showing a reflector
interactive display will not require any computations and will only involve swapping between memory buffers of various attributes. Keeping in view the size of seismic datasets, maintaining an array of memory buffers for all computed attributes is not possible. To overcome this problem, the library separates computed data into two types of outputs: fixed and floating memory buffers. Initially, the input trace, its computed
quadrature trace, and some base attributes like reflection strength and instantaneous frequency are computed and stored in four fixed memory buffers. These attributes are ready for display whenever required. Now, any derived attribute can be computed and stored into a floating memory, by using the precomputed data available in the fixed memory buffers. This saves some computation time and provides efficient
Fig. 6 Instantaneous frequency section indicating a gas reservoir highlighted by low frequency
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interactive display of the derived attribute. The floating memory is overwritten when a new derived attribute is computed. In this way, only limited memory buffers are required to efficiently display various attributes. Figure 4 shows an input seismic trace, its quadrature trace, and all instantaneous and wavelet attributes computed by this library. It can be seen that most of the instantaneous attributes are spiky and provide a higher resolution as compared to the blocky wavelet attributes. Figure 5 shows a seismic section with a prominent reflector, which is an interface between two layers. It can be seen that this section only provides structural geometry of subsurface layers. It does not provide any information regarding lithology, porosity, or fluid content. Figure 6 shows an instantaneous frequency attribute map of the seismic section given in Fig. 5, computed by this library. It can be seen that this attribute acts as a direct hydrocarbon indicator by highlighting a gas reservoir through drop in frequency. In this way, various combinations of attributes can be used to get useful geologic information such as lithology, porosity, fluid content, and other reservoir parameters.
Conclusions Seismic attributes provide useful geologic and reservoir information which is not obvious in the seismic section. Efficient display of various attributes involves successive computations over the whole seismic data volume, which require processing and memory resources. A digital signal processing library is presented for efficient computation of various post-stack instantaneous attributes as well as their derived wavelet and weighted average attributes, by optimizing computation time and memory. This library can be used in any seismic interpretation or visualization application for interactive display of various attributes. It will also be useful in multi-attribute analysis which involves simultaneous use of several attributes for reliable and robust interpretation.
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