3. ANALYTICAL TECHNIQUES
3.4. INDUCTIVELY COUPLED PLASMA-MASS SPECTROMETRY (ICP-MS)
I have proposed an antileakage method, the ALLSSA, to mitigate the spectral leakages in the least-square spectrum. I have shown the outstanding performance of the ALLSSA and compared it with the ALFT, ASFT, and the IMAP. The ALLSSA is a robust method that performs very well for analyzing data series and can be applied to non-stationary data series with more complex structure by an appropriate segmentation of the data series. The antileakage spectrum can estimate the exact location of the significant spectral peaks and consequently can be used to regularize unequally spaced data series. The ALLSSA is applied to analyze data series; however, it can be applied to analyze and regularize unequally spaced time series as well, where the spectrum is in term of frequency rather than wavenumber.
Simultaneous estimation of wavenumbers and amplitudes with constituents of known forms significantly reduces the number of iterations and improves the regularization results compared to the ALFT, ASFT, and the IMAP. Note that when the constituents of known forms and weight matrices are not considered in the iterative process, the ALLSSA is in fact the IMAP.
4 Applications of the Antileakage Least-Squares Spectral Analysis in Seismology
In this chapter, the ALLSSA is applied to regularize irregularly spaced synthetic 2D seismic data and 2D field seismic data given in time-distance (t-x ) domain that could be a crossline (inline) section of a 3D seismic data or a 2D seismic data acquired from a 2D seismic survey. The results are compared with the ALFT, ASFT, and nonlinear shaping regularization. Sections 4.2 and 4.3 are unabridged versions of sections
“SYNTHETIC DATA EXAMPLE” and “FIELD DATA APPLICATIONS” in Ghaderpour et al. (2018b), respectively.
4.1 Introduction to Marine Seismic Data Processing
In seismology, the seismic refraction23 method uses the refraction of seismic waves on geologic layer and rock/soil to characterize the geologic structure. This method is frequently used in engineering geology, exploration geophysics, etc. For instance, geophysicists apply this method to find oil associated with salt domes. Seismic waves generated by a vibrator or air gun (source) travel through the Earth, and their energies partially reflect and partially refract through the interface. Geophysicists reconstruct the pathways of the waves by measuring the traveling time from the source to several receivers and the velocity of seismic waves depending on the medium in which they are traveling. In marine seismic data acquisition, the receivers are hydrophones24 that convert sound waves25 (pressure waves) into electrical signals and record the pressure amplitude. In other words, hydrophone sensors measure the pressure wave field26. The response of each receiver to a single shot from the source is known as a “trace”, and it is recorded onto a data storage (generating a time series). As the vessel travels the shot location moves, and the process will be repeated.
23Seismic refraction is a geological principle based on the law of refraction (Snell’s law).
24A hydrophone is a type of microphone designed to measure the underwater sound.
25Sound waves tend to travel faster in denser media. The speed of sound in the air is about 344 m/s (20 degrees Celsius) and in the sea water is about 1531 m/s (20 degrees Celsius).
26The voltage generated by the hydrophone is proportional to the amplitude of the pressure wave as it passes by the hy-drophone (Hurrell, 2004).
Figure 4.1: 2D visualization of seismic survey.
Depending on the area of interest, the seismic images may contain many linear and/or curved events27. In traditional marine seismic survey, a vessel carries one or more cables (streamers) containing a series of hydrophones separated by a constant distance. The cables are deployed beneath the surface of the water and are positioned at a set distance away from the vessel and can be as long as six to eight kilometres (Figure 4.1). The 2D surveys use only one cable and one source; whereas, the 3D surveys usually use six or eight or more cables and sources. The inexpensive 2D surveys usually take place for general understanding of the regional geologic structure before drilling or performing 3D surveys. In 3D surveys, the term “inline” refers to the direction in which the receiver cables are deployed (the shooting direction), and the term “crossline”
refers to the direction that is perpendicular to the orientation of the receiver cables (parallel to the water surface). In other words, the inline is the direction of the vessel movement (parallel to receivers), and the crossline is perpendicular to vessel movement. The reader is referred to Scales (1997); Crawley (2000); Biondi (2006) for more details on seismic data acquisition and processing.
27In seismology the term “dip” refers to a slop angle of the foot-wall block measured clockwise from horizontal (0 to 90 degrees). A linear event is an event whose dip is constant. A curved event is an event whose dip gradually changes when observed at different times and offsets. A strongly curved event is an event whose dip changes over short ranges of offset (see also Chapter 2 for more detail).
Marine seismic data sets are usually irregularly sampled along spatial directions because of cable feath-ering28, editing bad traces, economy (acquiring more data is expensive and difficult because of current marine-acquisition technology), etc. They are more coarsely sampled along the crossline direction than the inline direction. Regularly sampled seismic data are required for various purposes including wave equation migration, seismic inversion, amplitude versus azimuth or offset analyses and surface-related multiple elim-ination (Weglein et al., 1997; Dragoset et al., 2010). An ideal way of obtaining regularly sampled data is to acquire more data. In 3D seismic surveys, acquiring more data may be done by increasing the num-ber of streamers with a wider azimuth range, but this is expensive and difficult to achieve, and so data regularization (reconstruction of data over any desired regular grids) becomes an important task.
The area spanned by a 3D seismic image is subdivided into small grids called stacking bins. Each trace in a 3D seismic data is positioned so that it passes vertically through the midpoint of each bin. The 3D seismic data can be viewed in terms of time-crossline-inline (t-x-y). The simplest 3D data regularization29is to bin the data and disregard the true data acquisition locations in the bin. This binning technique may cause artifacts that contaminate the final migrated images (Xu et al., 2005). Similar to the ALFT and ASFT, the true locations of the acquired data (using their latitudes and longitudes) may be used in the extension of the ALLSSA for regularization of 3D seismic data or higher orders (5D, inline-crossline-azimuth-offset-frequency) seismic data (Trad, 2009; Xu et al., 2010) that is subject to future work.
It is customary to transform each trace from the time domain to the frequency domain using the Fast Fourier Transform (FFT) (Spitz, 1991; Abma and Claerbout, 1995; Xu et al., 2010; Guo et al., 2015). Then for each frequency, generate a data series30whose data points, located at the trace locations, are the Fourier coefficients of that frequency (a temporal frequency slice). For irregularly spaced seismic data, each temporal frequency slice is an irregularly sampled data series, and the ALFT or ASFT or IMAP or ALLSSA method may be used to regularize the data series, and then each trace can be transformed back from the frequency domain to the time domain by taking the inverse FFT. Note that in this chapter and also the next chapter, all the analyses are performed on the temporal frequency slices.
In order to visualize the aliasing effects, the seismic data may be transformed from the t-x domain to frequency-wavenumber31 (f-k ) domain using the two-dimensional discrete Fourier transform (DFT). More precisely, suppose that {f (tj, x`) : 1 ≤ j ≤ m, 1 ≤ ` ≤ n} contains the values of the seismic data in the
28The lateral displacement of the cables is often called cable feathering. In the 3D surveys, due to sideways drift, the cables are not parallel to each other, and so the hydrophones are not regularly spaced.
29A popular method of 3D data regularization is using the least-squares collocation that is computationally slow for typical seismic data applications (Fomel, 2001, Chapter 2).
30In this chapter, the data series is obtained from 2D seismic images. In 3D seismic images, an irregular set of data points over irregular grids may be obtained.
31Frequency is the number of cycles per unit time, and wavenumber is the number of cycles per unit distance.
t-x domain, where tj = j/m and x`= `/n. The two-dimensional DFT for each pair (k1, k2) ∈ K1× K2 is defined as
f (kˆ 1, k2) =
m
X
j=1 n
X
`=1
f (tj, x`)e−2πik2x`e−2πik1tj, (4.1)
where K1= {−m2, . . . ,m2 − 1} and K2= {−n2, . . . ,n2− 1} if m and n are even and K1= {−m−12 , . . . ,m−12 } and K2= {−n−12 , . . . ,n−12 } if m and n are odd. Since the two dimensional DFT is defined for data sampled in regular grids, in the following examples, when a trace at spatial location x` is removed from the seismic image, the value of f (tj, x`) in Equation (4.1) is set to zero for all j’s. The f-k spectrum is defined by calculating the absolute values of ˆf (k1, k2) for all k1∈ K1and k2∈ K2.