0002

An Introduction to Seismic Interpretation

Chapter 2: Principles of the Seismic Method

2.1 Overview

Conventional refl ection seismic technology uses acoustic waves (“sound”) to image the subsurface. Conceptually, as shown in FIGURE 2.1, we begin by generating a “bang” at the earth’s surface. Following the bang, the sound travels down into the earth where some of it is refl ected at buried interfaces where there is a change in a physical property known as the acoustic impedance. As described in detail later in this chapter (SECTION 2.4) acoustic impedance is a function of a material’s P-wave velocity (i.e., how fast sound travels through the material) and its density. The acoustic energy refl ected at those interfaces is recorded at the surface as “echoes”. The distance from the surface to the buried horizons that generated the refl ections is measured in units of time – the two-way traveltime (TWT) being a measure of how long it took for the energy to pass from the surface source to the refl ecting horizon and return back to a receiver located at the surface. The strength (amplitude) of the refl ection is also recorded.

Usually we are interested in knowing how deep (in meters or feet) an interface is, rather than how long it takes for sound to travel to it and back to the surface. If we know the velocity of sound in the propagating medium we can derive true depths using a simple equation:

D = V x T [2.1]

Where D is the distance (depth) to a horizon, V is the average velocity and T is the time it takes for the sound to travel from the surface to the horizon (a one-way traveltime).

Most people have an intuitive understanding of how Equation 1 works. For example, if we know that it takes approximately 2.5 hours to drive between City A and City B, and we know that our average velocity is 120 km/h, we can easily calculate that the distance between those two points is approximately 300 km1. Likewise, suppose we know that the two-way traveltime to a refl ection is 2.302 seconds, and we know that the average velocity was 2022 m/s 2. We calculate the one-way

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1 Of course, our velocity may not have been constant for the entire length of the voyage. For example, we may havesped up or slowed down because of traffi c or (perhaps) the presence of law-enforcement offi cers.

2 Again, the velocity may be variable over the path traveled by the sound.

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traveltime as 1.151 seconds, and subsequently the depth to the horizon that generated the refl ection is 2327 m. This is the principle used during conventional bathymetric profi ling. A ship-borne source emits acoustic waves (“pings”) that travel through the water column, refl ect off the seafl oor, and are recorded at the surface. A constant water velocity (e.g., 1450 - 1500 m/s) is then used to convert the two-way traveltime to a water depth. In world of refl ection seismology, one of our challenges will be to determine how fast the sound has traveled in the subsurface between source and receiver because, as described below, velocity usually changes vertically (i.e. with depth) and laterally.

FIGURE 2.2 shows a simple example of how seismic imaging (i.e. vertical axis in units of time) can distort our view of the subsurface geology. In this example, the water depth changes from one side to the other. The velocity of sound in water in this instance is 1450 m/s, and the velocity of the rocks below the seafl oor is 2500 m/s. A horizontal surface, “Horizon A”, 500 m below the seafl oor appears to bow down to the right in the seismic data as the water gets deeper. Clearly the seismic image is not showing the true subsurface structure. The fundamental problem is that the seismic image shows how long it takes for sound to travel from the surface to a horizon and be refl ected back up to the surface. On the left, the relatively slow water is 50 m deep whereas on the right the water depth is 200 m. As such, it takes the sound longer for sound to travel down to Horizon A and back on the right than on the left. We will return to velocity-related imaging problems in later chapters.

In practice, refl ection seismology is more complicated than the simple cases described above. The optimal source of acoustic energy for the study being performed needs to be determined, there are many interfaces in the subsurface that generate refl ections, velocities change laterally and vertically, and we need to fi nd cost-effective ways to collect 2-D seismic profi les or 3-D seismic volumes. Practical aspects of seismic acquisition and processing are described in CHAPTER 3. In this chapter the focus is on the physical basis of the seismic method, including topics such as the controls on the velocity at which sound propagates, the strength of the refl ection, and seismic resolution.We can represent the paths of the energy generated by our

An Introduction to Seismic Interpretation • Chapter Two— Principles of the Seismic Method

FIGURE 2.1:

Schematic representation of how sound is used to image the subsurface

FIGURE 2.2:Simple example of how seismic imaging can distort our view of the subsurface geology


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bang in two different ways (Figures 2.1, 2.3). The fi rst is to depict a wavefront that expands out from the energy source. A commonly used analog is to consider the ripples that spread out on the surface of a pond when a pebble is dropped into the water. The ripples are the wavefront that expands away from the energy source. A second way to depict the expansion of energy away from the source is to show raypaths that connect the source point with the wavefront, i.e. the direction of energy propagation. If the velocity of wave propagation is the same in all directions, the wavefront spreads out in a spherical form and the raypaths are straight as shown in FIGURE 2.3A. When the velocity varies in different directions (the usual case in the real earth), the raypaths bend and the wavefront is no longer spherical, as shown schematically in FIGURE 2.3B. It is common practice to illustrate concepts of refl ection seismology using raypaths, partly because the mathematics of raypaths are simpler than those of propagating wavefronts, but also because raypaths are easier to draw.

2.2 Characteristics of Waves

There are several types of waves that can propagate through the Earth or at its surface. As noted in the PREVIOUS SECTION, most seismic datasets are collected using acoustic waves, also known as “compressional” waves, “P-waves”, or simply sound waves. Another type of wave used in refl ection seismology is a shear wave, also known as an “S-wave”. Most geologists are introduced to compressional and shear waves in the context of earthquakes. Compressional waves are referred to as primary or P-waves in earthquake seismology because they are the fi rst waves recorded by a seismograph following an earthquake. The shear waves are the second waves (“secondary waves”) recorded by a seismograph. Because of the differences in traveltime, it should be clear that compressional waves travel at a higher velocity than shear waves. Furthermore, shear waves cannot propagate through fl uids (this is one of the lines of evidence that has been used by geophysicists to determine that the earth’s outer core is in a liquid state). Other types of waves (e.g., surface waves known as Rayleigh waves) are also generated by earthquakes and in refl ection seismology, however we will not discuss them here.

FIGURE 2.4 shows the particle motions induced in the earth as P-waves and S-waves propagate through it. The


FIGURE 2.3:

Raypaths connect the source point with the wavefront, depicting the expansion of energy away from the source

FIGURE 2.4:Representation of particle deformation during the passage of a compressional wave (P-wave) and a shear wave (S-wave)


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passage of a compressional wave causes a portion of rock to alternately contract and then dilate. The rock body changes shape and volume. On the other hand, the shearing motions associated with the passage of a shear wave cause changes in shape, but not in volume.

This text focuses primarily on the use of P-wave seismic methods because they are the most common type of wave used in refl ection seismology. It should be noted however that: a) S-wave seismic methods are seeing increasing use, and these uses will be discussed later (CHAPTER 8), and b) the geologic interpretation of S-wave seismic data has many similarities to P-wave interpretation. P-wave and S-wave seismic data are sometimes recorded together as “multicomponent” seismic data.

Waves, seismic or otherwise, can be described in various ways. A simple cosine, although not necessarily representative of an actual wave used in refl ection seismology, can be used to help defi ne these terms (FIGURE 2.5). The wave consists of changes in amplitude that can be represented by a series of positive and negative values. A peak represents a local maximum amplitude value and a trough represents a local minimum value. As seismic waves travel through the earth, they have a physical length (meters or feet; FIGURE 2.5A). The wavelength is the distance between successive repetitions of the waveform (e.g., the distance between two successive troughs or two successive peaks). Note that the cosine curve shown in the fi gure is a repeating waveform.

A different way of showing a wave is to show how amplitudes change at a given point through time. For example, imagine that you are standing in the surf zone of a beach, and that the waves generated offshore pass you on their way to the beach. You could record changes in the water surface elevation as a function of time. If the waves are cosine curves (an unlikely situation in a surf zone), then the plot of amplitude (water surface elevation) versus time might look like the one shown in FIGURE 2.5B. In this case, the separation (in time) between successive wave crests is known as the wave period (T).

The key variables for defi ning a wave’s shape are its amplitude, frequency and phase. For example, the equation for a cosine curve can be expressed:


FIGURE 2.5:

A simple cosine representing traveling waves in distance (A) and time (B)

FIGURE 2.6:Examples showing the effects of changing the amplitude, frequency and phase of a cosine curve


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f(x) = a × cos(bx + c) [2.2]

where: a controls the amplitude of the curve, b controls the frequency, and c the phase of the curve. Examples showing the effects of changing the amplitude, frequency and phase of a cosine curve are shown in FIGURE 2.6. The cosine curves shown in FIGURE 2.5 are monochromatic, i.e. they consist of only one frequency.

Frequency Content of Seismic Data

The frequency (F) of a wave train is the number of cycles (e.g., “peaks”) that pass by a given point in a given time. Frequency is measured in cycles/second or Hertz (abbreviated as Hz). For example, 40 Hz describes a case where 40 cycles (e.g., peaks) will pass by a given point in one second. The frequency can be expressed as the inverse of the period:

TF

1 [2.3]

For example, a period of 0.05 seconds corresponds to a frequency of 20 Hz. Typical petroleum-industry seismic data have frequencies that are in the 10s of Hertz range (e.g., 10 – 80 Hz). Shallow land surveys might have data in the 100s of Hz range, whereas some high-resolution marine profi ling systems use frequencies that extend up into the kHz range.

A key relationship for seismic interpreters relates the wavelength (λ) to the wave propagation velocity and the frequency:

FV / [2.4]

Two examples show how this equation might be used. Assume that the P-wave velocity is 5000 m/s and that the frequency is 50 Hz. The wavelength will be 5000/50 = 100 m. Now assume that the velocity remains 5000 m/s but the frequency is now 25 Hz. The wavelength will be 5000/25 = 400 m. As discussed below, the wavelength places fundamental limits on seismic resolution and the limit of seismic detectability.

Seismic wavelets do not repeat like the cosine curves presented in FIGURE 2.5. The simplest way of thinking about a seismic wavelet is to consider it to be the acoustic pulse that was generated by the seismic source (FIGURE 2.1). Simm and White (2002) discuss various aspects of seismic wavelets. A more realistic seismic wavelet, but still highly simplifi ed, is shown in FIGURE 2.7A.Note that this wavelet does not repeat itself. Furthermore, this wavelet contains a range of frequencies, rather than just one like the monochromatic waves of FIGURE 2.5. FIGURE 2.7B shows how several frequencies can be combined to generate a wavelet such as the one shown in part A of that fi gure. Four different cosine curves are shown in FIGURE 2.7B, all four having identical amplitudes, but different frequencies. Additionally, in all four cases the location of the central peak amplitude is the same (i.e., they all have the same phase). Summation of those four curves produced the wavelet shown in part A. In the middle of the wavelet, the positive values add constructively to generate a strong central peak. On either side of that peak, positive and negative values are being summed, and they tend to cancel each other out, especially as the distance from the central peak increases. Non-perfect cancelling leads to the generation of side lobes. FIGURE 2.7B only shows four discrete frequencies being added. In practice, the broader the range of frequencies being added



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(i.e., the broader the bandwidth), the sharper the central peak will be and the smaller the side lobes. The importance of reducing the side lobes will be discussed later.

The world of music provides an analogy for how a wavelet might contain a range of frequencies. Imagine a small child playing with a piano. Using one fi nger, the child can play one note (frequency) at a time, including high notes (high frequencies), low notes (low frequencies) and notes (frequencies) in the middle. Then the child uses her forearm to smash on 15 piano keys at once. The child has produced a wave containing several frequencies that sounds more like a “bang” than an individual note. In the same way, the bang produced by an explosive source (airgun, dynamite, etc.) creates a wavelet that consists of a broad range of frequencies (e.g., 10 – 100 Hz).

As shown by EQUATION 2.4, the frequency content of the seismic data partially controls (along with the velocity) the wavelength. Although it is possible to directly measure the frequency content of the source pulse, this is not routinely done (except for vibroseis sources; see CHAPTER 3). Furthermore, and for reasons described below, the frequency content of seismic data generally varies with depth (time). Interpreters therefore seek to determine the frequency content of a portion of seismic data in a variety of ways.

One method requires the interpreter to defi ne a portion of the seismic data for the analysis (e.g., a subset of the seismic data defi ned by a range of two-way travel times and areal extent; some seismic interpretation software packages allow the user to choose this window by clicking and dragging the mouse to defi ne an analysis window). A Fourier transform (FT) can be used to convert the input data (amplitude versus time) into an amplitude spectrum (amplitude versus frequency) such as the two examples shown in FIGURE 2.8. The x-axis shows the frequency (in Hz) and the y-axis shows a measure of the relative strength of the signal at that frequency (commonly in either decibels or power).

The amplitude spectrum in FIGURE 2.8A is from some seismic data acquired on land using a dynamite source. It shows a broad range of frequencies that extends from approximately 10 to 100 Hz. The apparent presence of


FIGURE 2.7:

Simplifi ed wavelets representing various aspects of seismic data

FIGURE 2.8:Amplitude spectra for two different seismic data sets (from two different seismicinterpretation packages) showing the range of frequencies in the data


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frequencies below 8 or 10 Hz is a mathematical artefact (those frequencies are generally either not recorded during data acquisition or may be fi ltered out during processing to remove a surface wave called ground roll) as is the presence of frequencies above ~100 Hz. The amplitude spectrum in FIGURE 2.8B is from a different land 3-D seismic data set that has a much more restricted range of useable frequencies, extending from approximately 15 – 40 Hz. Note too that the strength of the signal for different frequencies is quite variable in that range3. In this case, frequencies above 40 Hz were digitally removed (fi ltered out) during data processing in an attempt to reduce noise (i.e., processors thought that frequencies recorded above this level were primarily related to noise, not useful seismic refl ections).

Because the seismic wavelet contains a range of frequencies, rather than a single frequency, we need to be able to characterize it using a single dominant frequency in order to calculate the wavelength. As shown in FIGURE 2.7A, the dominant frequency is represented by the separation in time between the fi rst side lobes. This measure was called the “wavelet breadth” by Kallweit and Wood (1982). The dominant frequency is used to calculate the wavelength. One common way to estimate the dominant frequency from an amplitude spectrum is to take the peak value on an amplitude spectrum (Liner, 2004). Alternatively, if the amplitude spectrum is relatively fl at, the maximum and minimum useable frequencies (referred to as the upper terminal frequency – fu - and lower terminal frequency – fl – respectively by Kallweit and Wood, 1982, also known as low-pass and high-pass frequencies, respectively) are estimated, and the dominant frequency is given as the midpoint between those two values.FIGURE 2.9A shows an idealized amplitude spectrum and how the dominant frequency would be derived from it. FIGURE 2.9B shows a real amplitude spectrum. The defi nition of the dominant frequency becomes somewhat more ambiguous in this case, because the midway point between the low- and high-pass frequencies (approximately 35 Hz) does not correspond to the maximum signal strength (approximately 42 Hz).

zHsm

cyclesxsmcyclespeaks

83

100083

0101

100)(8.3


FIGURE 2.9:

Comparison of an idealized amplitude spectrum and a real amplitude spectrum

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3 In the piano case, this would mean that the child is hitting some of the notes harder than others.


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Another way to estimate, this time by inspection, the dominant frequency of some seismic data is to use a procedure known as counting peaks. This “quick-look” method does not require us to mathematically transform the seismic data to the frequency domain via a Fourier transform. Instead, the interpreter identifi es a narrow time interval in the data, perhaps 100 ms, and counts the number of cycles (e.g., peaks) in that interval. The portion of seismic data chosen for this analysis should be relatively noise free and consist of relatively equally spaced peaks. If a 100 ms time window is used, the dominant frequency of the seismic data in that interval can be estimated by multiplying that number by 10. For example, suppose 3.8 peaks are counted in a 100 ms interval at a location below the red arrow in FIGURE 2.10 (the number of peaks will not always be a whole number). We can convert this count to an equivalent frequency by multiplying the numerator and denominator by 10:

zHsm

cyclesxsmcyclespeaks

83

100083

0101

100)(8.3

Finally, we might identify a single refl ection in the data and carefully measure the distance (in two-way traveltime) between two successive peaks (or troughs). This will be an estimate of the period at that point in the data. With this information we can calculate the dominant frequency using EQUATION 2.3. FIGURE 2.11A shows how this might be done. The time separation between two peaks at the red arrow in the image can be measured as approximately 14 ms. This estimate of the period can be used to estimate the frequency at that level:

zHT

F 17014.011

An amplitude spectrum derived from approximately the same level is shown in FIGURE 2.11B. Note that the 71 Hz frequency derived from estimating the period is high, when compared to the dominant frequency (approximately 50 Hz) that might be estimated using the maximum and minimum useable frequencies, although it is closer to the dominant frequency that would be estimated by picking the peak amplitude in the spectrum. This is because the spacing between two peaks is actually controlled by the interaction between two variables: the seismic frequencies and the spacing between beds. As such, estimates of frequency content derived from too a narrow an analysis


FIGURE 2.10:

Counting peaks to estimate the dominant frequency

FIGURE 2.11:Measuring wave period to estimate the dominant frequency


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window (e.g., measuring the period of a single refl ection) may not be representative of the true frequency content of the data.

These reservations aside, with time and practice, an experienced interpreter can estimate the dominant frequency of some seismic data visually with approximately the same accuracy as can be derived from an amplitude spectrum. However, the amplitude spectrum has an advantage in that it provides information about the range of frequencies in the data. From a practical perspective, the dominant frequency estimated using any of these methods is exactly that – an estimate. As such, it might be more honest to give a possible range of values (e.g., “FD is between 25 and 30 Hz”) than to assume that an exact value (e.g., 27.3 Hz) is more accurate.

The range of frequencies embedded in the seismic wavelet is known as the bandwidth. Ideally, the seismic wavelet contains a broad range of frequencies, including both low and high frequencies. In this way, the pulse is sharper (i.e. narrower) and wavelet sidelobes are reduced, as shown in FIGURE 2.12. The useable bandwidth is typically defi ned in Hz using the low- and high-pass frequencies (e.g., 10-90 Hz).

The wavelets used in FIGURE 2.12 are known as bandpass wavelets. They are constructed mathematically, using frequency fi lters, and hopefully they resemble the wavelets that are embedded in our seismic data. Butterworth and Ormsby wavelets are bandpass wavelets that are named after the mathematicians who defi ned them. This type of wavelet is defi ned using four frequencies, known as the low-cut, low-pass, high-pass and high-cut frequencies, as shown in FIGURE 2.9A. No energy is present below the low-cut frequency or above the high-cut frequency. All frequencies between the low-pass and high-pass frequencies have equal amplitude in one of these wavelets (although this is not necessarily true in actual seismic data, e.g., FIGURE 2.8B), and amplitudes ramp up between the low-cut and low-pass frequencies, and ramp down between the high-pass and the high-cut frequencies.

Seismic Phase

The phase of the seismic wavelet is also a key variable to a seismic interpreter. The effects of changing the phase


FIGURE 2.12:

Effects of changing the frequency content of a wavelet on wavelet shape


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of a cosine curve were displayed in FIGURE 2.6. For a seismic wavelet, let us begin by considering two different types of phase.

A minimum phase wavelet (FIGURE 2.13A) approximates the type of wavelet generated by an airgun or dynamite energy source. It is called a minimum phase wavelet because the highest amplitude has a “minimum delay” from the beginning of the wavelet (the “reference point”, time = 0, at which we begin to generate the wave). After that the amplitude oscillates a few times then it dies away to nothing (Liner, 2004). Consider a bell when struck by a hammer. The amplitude (volume) of the signal emitted by the bell builds up rapidly as the hammer begins to hit it (t = 0), eventually reaching a maximum. The bell rings after the impact, with the amplitude of the ringing diminishing with time.

Although minimum phase wavelets are wavelets that can be produced using seismic sources, they have disadvantages during the interpretation stage. One such disadvantage is the presence of the oscillations following the initial amplitude maximum. A refl ection generated from a single interface using a minimum phase wavelet might consist of a peak (an amplitude maximum) followed by a series of smaller peaks (FIGURE 2.14A). During the interpretation phase, it might be tempting to interpret the trailing peaks as weaker amplitude refl ections. Alternatively, they might interfere with refl ections from underlying horizons. A second disadvantage of a minimum phase wavelet is that the delay between the maximum amplitude and the onset of the wavelet (i.e. the reference point) is variable, and depends on the frequency content of the wavelet. The delay is greater for a low-frequency minimum phase wavelet than for a high-frequency minimum phase wavelet (FIGURE 2.15). From an interpretation perspective, we would like the maximum amplitude to coincide with the interface that generates the refl ection. The delay, and variability of the delay with frequency content, adds unwanted ambiguity about where the interface truly is.

In contrast, a zero phase wavelet has a maximum, or a minimum, amplitude (peak or trough respectively) that is centered on the reference point (FIGURE 2.13B). From an interpreter’s perspective, this means that the interface generating the refl ection should correspond to the maximum (or minimum) amplitude value of the refl ection


FIGURE 2.13:

Comparison of a minimum-phase wavelet with a zero-phase wavelet that has the same frequency content

FIGURE 2.14:Comparison of the seismic response of a minimum-phasewavelet (A) and a zero-phase wavelet (B) to a single interface

FIGURE 2.15:Comparison of delays in a low-frequency minimum phase wavelet and a high-frequency minimum phase wavelet


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(FIGURE 2.14B), providing that interference from adjacent beds can be neglected (see below). Furthermore, this will be true regardless of the frequency content of the wavelet (FIGURE 2.16). As such, the interpreter can map an interface with greater confi dence. A problem with zero phase wavelets is that they cannot be produced in nature. Consider the zero phase wavelet shown in FIGURE 2.13B. Amplitudes change before the reference point (t = 0), which would be akin to our bell making a sound before it is struck. Clearly this is impossible. In practice, zero phase wavelets are generated mathematically during a processing step called deconvolution. Discussion of this processing step will be deferred until the NEXT CHAPTER.

A third type of wavelet would be one in which the phase has been rotated away from zero (possible range of rotation between ± 180°). FIGURE 2.17 shows four different wavelets, each with the same range of frequencies but different phases. Note that the symmetrical shape of the wavelet at 0° changes as the phase is rotated. The wavelet used in this example is known as a Ricker wavelet. A zero-phase Ricker wavelet consists of a peak that is fl anked, symmetrically, by a single pair of sidelobes. The amplitude spectrum in the lower part of that fi gure shows that, unlike bandpass wavelets, Ricker wavelets are characterized by a single peak frequency value. Ricker wavelets are commonly used during seismic modeling, because of their simplicity, even though they do not capture the full complexity (e.g., number of side lobes) of real seismic wavelets.

In the same way that it is important for an interpreter to determine the frequency content of a seismic dataset, it is also important to determine the data’s phase. Acquisition and processing parameters that affect phase are discussed in CHAPTER 3, the concept of data “polarity” is discussed in CHAPTER 4, and means for quantitatively determining data polarity are discussed in CHAPTER 5. Further discussion of phase is deferred to these chapters.

2.3 Rock Properties

We now turn our attention to the factors that control rock (or sediment) physical properties. As mentioned at the outset of this chapter, and discussed in detail below in SECTION 2.4, the primary physical property of interest for P-wave seismic methods is the acoustic impedance (Z),


FIGURE 2.16:

Effect of frequency content on positioning of a zero-phase wavelet

FIGURE 2.17:Effects of changing wavelet phase on the shape of the wavelet


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the product of a material’s P-wave velocity (α) and its bulk density (b)4:

bZ [2.5]

In turn, and as described in this section, velocity and density are controlled by variables such as porosity, mineralogy and pressure. These topics are covered in greater detail, and with greater physical rigor, by Sheriff and Geldart (1995), Mavko et al. (1998), Liner (2004), Schön (2004), Avseth et al. (2005) and others. For the purposes of this text, where the focus is on the geologic interpretation of seismic data, the treatment will not be in as much detail.

Elastic Properties and Velocities

SECTION 2.2 discussed P-waves and S-waves. As shown in FIGURE 2.4A, the passage of a P-wave induces changes in the volume and shape of a body. FIGURE 2.4B shows that the passage of an S-wave induces changes in the shape of a body, but not its volume. In principle, the body returns to its original shape and volume following the passage of the wave. In both cases, the body is subjected to a stress (e.g., compressive stress and/or shear stress) and undergoes strain (i.e., deforms) and then recovers and returns to its original shape after the stress ceases. The amount of strain induced in a body by a given amount of stress is controlled by properties known as the elastic modulii. We focus here on two of these modulii, the bulk modulus and the shear modulus.

The bulk modulus (κ) is a measure of the compressibility of a body. It is the stress-strain ratio under simple hydrostatic pressure, and measures the body’s propensity to change volume, not shape. The bulk modulus is the inverse of the compressibility (e.g., the more compressible a body is, the lower the bulk modulus) and so it is sometimes called the “incompressibility”. The term “hydrostatic pressure” in the defi nition above means that the body is squeezed equally in all directions. If we take a sample in the laboratory and squeeze it this way (FIG. 2.18A), we can defi ne the bulk modulus via the following measured properties:

VPV

[2.6]

where ΔP is the measured change in pressure, ΔV is the measured change in volume, and V is the original volume. Inspection of this equation shows that the larger the change in volume for a given pressure (i.e, the more compressible the body), the smaller the bulk modulus will be.

In principle, the bulk modulus can be broken down into three components, one related to the incompressibility of the mineral grains (e.g., quartz sand grains), one related to the framework (e.g., how well cemented or compacted the rock is), and the third related to the pore-fi lling fl uids (water being incompressible, gas being compressible). Avseth et al. (2005) suggested that the bulk modulus can be approximated using the following expression:

mineralrock

11 [2.7]

where κrock is the bulk modulus of the saturated rock, κmineral is the bulk modulus of the minerals


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4 Units of acoustic impedance are strange, and might be meters per second x kilograms per meter cubed, or perhaps feet per second x grams per cubic centimeter, depending on the input values used to measure velocity and density.


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comprising the rock, ø is the porosity and κø is the bulk modulus of the saturated pore space. This formulation emphasizes that the rock’s bulk modulus is related not only to the mineralogy but also to the porosity and the nature of the pore-fi lling fl uids. Note the inverse relationship between bulk modulus and porosity suggested by EQUATION 2.7. All else being equal, the higher the porosity, the more compressible a rock will be.

The shear modulus (μ), also known as the rigidity, is the stress-strain ratio for simple shear, and measures a body’s ability to resist shear deformation. It is associated with a change in shape not volume. To measure the shear modulus of a body in the laboratory, we might take our sample and apply a shearing stress (FIGURE 2.18B). The shear modulus will be defi ned using:

lxAF/

/

[2.8]

where F is the force, A is the area on which the force acts (F/A is the shear stress), Δx is the transverse displacement, and L is the initial length (Δx/l is the shear strain). An important point to note about the shear modulus is that its value is unaffected by the nature of any pore-fi lling fl uids. A gas-fi lled sand will have the same shear modulus as an equivalent sand (mineralogy, cementation, porosity, etc.) having its pores fi lled with water, or oil, or any combination of these fl uids. This is because fl uids have no shear strength (no rigidity), and therefore they do not add to the rigidity of the matrix.

The equations for the bulk and shear modulii presented above assume two things. First, there is no permanent distortion of the sample (i.e., the deformation is truly elastic). Second, the equations assume that the material is isotropic. This second assumption might mean, for example, that it is not easier to squeeze a rock parallel to bedding than perpendicular to bedding. Anisotropy can be important in some rocks, for example shales or fractured rocks. Because physical properties are not uniform in all directions, the elastic modulii (and hence velocities, as described below) might be different parallel versus perpendicular to bedding (shale), or parallel versus perpendicular to fractures.


FIGURE 2.18:

Schematic illustration of the types of strain measuredby the bulk modulus (κ) and the shear modulus (μ)

FIGURE 2.19:Plot of experimentally measured P-wave velocity (α) versus density (b) for various sedimentary rock types


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An additional property that we need to consider is the bulk density of a rock or sediment. Density is a mass per unit volume (e.g., g/cm3 or kg/m3). The bulk density of a rock (ρb) is related to the density of the mineral matrix (ρma), the porosity (ø) and the density of the pore-fi lling fl uids (ρf) via the following mass-balance relationship:

ρb = ø × ρf + (1 - ø) × ρma [2.9]

Values of density for some common minerals and fl uids are listed in TABLE 2.1. Ji et al. (2002) and Schön (2004) presented a more comprehensive list of mineral and fl uid densities.

Note how, like the bulk modulus, the bulk density is a function of the mineralogy, the porosity and the nature of the pore-fi lling fl uids. Because the density of fl uids (ρf) is lower than the density of the mineral constituents (ρma), higher porosity rocks will have lower densities than lower porosity rocks of the same lithology. Similarly, a gas-charged sandstone will have a lower density than an equivalent sandstone that is “wet” (i.e. pore spaces fi lled with water) because the pore-fi lling gas is less dense than water.

Mineral/Fluid P-Wave Velocity (m/s) Density (g/cc)Quartz 5490 2.65Calcite 6710 2.71Dolomite 7010 2.875Anhydrite 6100 2.98Gypsum 5790 2.35Salt 4570 2.03Water (Pure) 1400 1

Table 2.1 Compressional wave velocity and density for common materials in sedimentary deposits.

The main reason for our interest in these physical properties is that they control P- and S-wave velocities. The relationships between elastic modulii and rock velocity (P-wave and S-wave) were developed by Gassman (1951) and Biot (1956) and are commonly simply referred to as the “Gassman equations”. There are different formulations for relating elastic modulii to velocity, but a simple equation for predicting P-wave velocity (α) as a function of the bulk modulus, shear modulus and bulk density can be formulated as:

)3

(4 [2.10]

Inspection of this equation shows several things. First, the P-wave velocity is proportional to the bulk modulus. The harder it is to compress a rock, the higher the P-wave velocity. Second, the P-wave velocity is also proportional to the shear modulus. Third, although there appears to be an inverse relationship between bulk density and velocity, the same factors that control bulk density also control the bulk modulus. For example, increasing the porosity decreases the bulk density and the bulk modulus. As such, there is a positive correlation between P-wave velocity and bulk density,



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as noted experimentally by Gardner et al. (1974; FIGURE 2.19).

The relationship between velocity and density for most of the sedimentary rock types shown in FIGURE 2.19 can be approximated using the following empirical equation, commonly referred to as the Gardner Equation:

4)32.0

(

b [2.11]

The constants in this equation were derived to fi t the typical reservoir rocks (sandstone, shale, limestone, dolomite) in that fi gure and were derived when velocity units are in feet per second and density is expressed in grams per cubic centimeter. The constant in the denominator of the fraction on the right hand side is 0.31 (instead of 0.23) when velocity is in m/s. If velocity and density logs are available, the constants can be adjusted to give a better fi t for a particular rock type in a particular basin.

Another equation that is sometimes used, primarily in log analysis, to relate porosity and P-wave velocity of a rock is the time-average equation (Wyllie et al. 1956):

amf VV

)1(1

[2.12]

where Vf is the velocity of the pore-fi lling fl uids and Vma is the velocity of the rock matrix (see TABLE 2.1 for typical values; Schön, 2004, presented a more comprehensive list of mineral velocities). This equation suggests that the travel time of the acoustic signal through the rock is the sum of the partial travel time through the solid matrix and the partial travel time through pore space fi lled by a pore fl uid. Although this equation appears to be physically based, it is in fact empirically derived and its general applicability is questioned in seismic studies (e.g., Mavko et al., 1998; Eberli et al., 2003; Saleh and Castagna, 2003; Avseth et al., 2005). Gardner et al. (1974) however indicated that the time-average equation is adequate for relatively deeply buried brine-fi lled sedimentary rocks where the infl uence of pressure on velocity is minimal. In these cases velocity is essentially a function of porosity and mineralogy. There is however a general consensus that EQUATION 2.12is not adequate for predicting how changes in the composition of pore-fi lling fl uids will affect seismic response (see discussion of time-lapse seismic methods in CHAPTER 8).

The S-wave velocity (β) can be defi ned as:

[2.13]

Note that the S-wave velocity does not depend on the bulk modulus because there is no compression involved in the propagation of an S-wave (FIGURE 2.4). Note also that, because the shear modulus is unaffected by the nature of pore-fi lling fl uids, the effect of changing fl uids is only felt through the density term. As such, pore fi lling fl uids have a minimal impact on shear wave velocity.

TABLE 2.2 illustrates qualitatively how changing various physical parameters (temperature, overburden pressure, pore pressure, porosity, clay content and gas saturation) will affect the P-



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and S-wave velocities and density of a sedimentary deposit. Note that many of these parameters (porosity, pore-fi lling fl uids, lithology) are of direct interest to the petroleum industry. These relationships illustrate why refl ection seismology has become such a staple of that industry – the method responds to factors that are of direct interest.

P-Wave Velocity

S-Wave Velocity Density Bulk Modulus

Shear Modulus

With Increasing: Temperature Decreases Slight Decrease Unchanged Decreases Slight DecreasePressure Increases Increases Increases Increases IncreasesPore Pressure Decreases Slight Decrease Unchanged Increases DecreasesPorosity Decreases Decreases Decreases Decreases DecreasesClay Content Decreases Decreases Unchanged Decreases DecreasesGas Saturation Decreases Slight Increase Decreases Decreases Unchanged

Table 2.2 Changes in rock properties with changes in reservoir conditions. Compiled from various sources.

Changes with Burial and Diagenesis

FIGURE 2.20 illustrates how diagenesis (i.e. porosity reduction due to cementation, compaction and pressure solution) and sorting (including clay content) affect P-wave velocity. In that fi gure, the Suspension Line represents freshly deposited sediments of different sorting. Well-sorted sands have an initial porosity of approximately 40% (lower right) and more poorly sorted deposits have lower porosities. Porosity reduction through diagenesis causes the velocity to increase along the red lines for a given initial porosity. Porosity reduction through cementation and other processes increases the rigidity and incompressibility of the rock, causing the velocity to increase. The red line joining the “mineral point” (zero porosity) with the “clean, well-sorted sand point” represents a theoretical upper bound to velocity. The magenta lines show the trends for sediments of similar states of diagenesis (i.e., age) but different sorting (clay content). Increasing the clay content (not shown on the graph) “softens” the rock (clay has lower elastic modulii than quartz) resulting in a slower velocity for shaly sandstones than for clean sandstones for any given porosity.

Recall from TABLE 2.2 how physical properties such as porosity control velocity and density. Porosity tends to decrease with depth due to the combined effects of compaction and cementation. Inasmuch as velocity, density and acoustic impedance are inversely related to porosity, this means that these three variables typically increase with depth5. Sands and shales typically follow different compaction trends in sedimentary basins. FIGURE 2.21 shows typical acoustic impedance trends for these lithologies in a thick (few km), normally compacting sedimentary basin. In the shallow part of the section the acoustic impedance of unconsolidated, but somewhat cohesive mud is higher than that of the wholly unconsolidated sands. Mud compacts earlier in response to burial than sand, explaining why their impedance is higher than sand at shallow burial depths. With depth, both the sands and shales compact and so their velocities and densities increase, but at different rates. At a certain point in the burial profi le, the acoustic impedances of the two lithologies may be approximately equal. Finally, at depth the acoustic impedance of the lithifi ed sandstone is greater than that of the shale. This discussion assumes that the pore space of the sand is entirely fi lled with


__________________________________________________________________

5 Although obviously overburden pressure, lithology, temperature and other variables affect velocity too, as per TABLE 2.2.


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water. If part of that pore space is fi lled with hydrocarbons, different acoustic impedance relationships might apply (see CHAPTER 8).

The preceding discussion has focused primarily on siliciclastic deposits. Carbonate diagenesis differs from clastic diagenesis in many respects (e.g., Choquette and Pray, 1970). As just noted, porosity generally decreases with depth in clastic systems because compaction is driven by the weight of the overlying sediments. Simple relationships between porosity and depth may be absent in carbonate systems because early cementation (sometimes within months of deposition) may occlude porosity at shallow depths and/or inhibit subsequent burial compaction. Carbonate rocks and sediments are also different from siliciclastic deposits in that whereas their elastic properties are related to porosity, the type of porosity plays a signifi cant role in controling rock velocity in carbonates. As such, it becomes important to understand carbonate diagenesis, how diagenetic processes vary with depth, and how diagenesis affects rock properties in carbonates. For example, carbonate sediments can become cemented shortly after deposition and, in this case, the loss of porosity with depth due to mechanical compaction may be hindered. Carbonate minerals are also more succeptible to dissolution (secondary porosity development) and precipitation than silicate minerals such as quartz.

Anselmetti and Eberli (1997), Wang (1997), Eberli et al. (2003), Fournier and Borgomano (2007), and Weger et al. (2009) showed how the porosity fraction, pore type (e.g., moldic, intercrystalline) and cement type affect velocity in carbonate rocks. Differences in mineralogy (e.g., calcite versus dolomite) have a relatively minor infl uence on velocity. FIGURE 2.22 summarizes how diagenesis affects the p-wave velocity of carbonates. Cementation and compaction increase acoustic impedance, whereas dissolution initially has a relatively minor infl uence because the development of pores does not signifi cantly affect the rigidity or compressibility of the rock. Sucrosic dolomitization (a fabric-destroying process) increases porosity and decreases velocity.

Igneous and Metamorphic Rock Properties

Rock properties of igneous and metamorphic rocks were discussed by Salisbury et al. (1996), Milkereit and

FIGURE 2.20:

Generalized model for relationship between P-wave velocity (α) and porosity (Φ) in clastic deposits

FIGURE 2.22:Summary diagram illustrating how diagenetic processes affect relationships between P-wave velocity (α), porosity (Φ) and acoustic impedance trends of carbonates

FIGURE 2.21:Schematic representation of compaction trends for brine-fi lled (“wet”) sands and shale in a normally compacting sedimentary basin


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Eaton (1998) and Schön (2004). FIGURE 2.23 compares velocity-density trends for igneous and sedimentary rocks. Crystalline silicate rocks extend trends seen in sedimentary rocks to higher velocities and densities along a “Nafe-Drake” curve (Ludwig et al., 1970) that relates α to b for velocities ranging from 1.5 to 8.5 km/s. Christensen and Mooney (1995) also presented a relationship between α and b for crystalline rocks. Ultramafi c rocks have higher acoustic impedance than mafi c or felsic igneous rocks. Comparison of Figures 2.19 and 2.23 shows overlap in acoustic impedance between carbonates and felsic igneous rocks (velocities of 6,000 to 7,000 m/s and densities of 2.7 to 2.8 g/cm3). Note the relative position of sulphide and iron-oxide minerals. These ores have acoustic impedances that are signifi cantly in excess of most crustal rocks, with much of the difference being related to differences in density (mineral densities ranging from 4.1 to 7.6 g/cm3).

Rock Properties and Lithology - Summary

Figures 2.19, 2.22 and 2.23 all show lines of equal acoustic impedance superimposed on velocity-density or velocity-porosity relationships. From those fi gures, we note that: 1) Different types of (sedimentary) rock can have acoustic impedances that are different to, or the same as, the acoustic impedance of other types of rocks. As a general rule though, in most settings different rock types have different acoustic impedances. 2) A single rock type (e.g., “sandstone”, “shale”) can have a range of acoustic impedances. This should not be surprising because of, for example, variability in the porosity, mineralogy, pore-fi lling fl uids, burial depth (pressure) and other variables for “sandstones” or other lithologies.

2.4 Waves and Interfaces

SECTION 2.1 discussed raypaths and wavefronts and stated that, if only for simplicity, it is common to draw raypaths when representing seismic waves. SECTION 2.3, established the controls on P- and S-wave velocity. We have not yet discussed why refl ections occur. This section examines what happens when downgoing seismic energy encounters an interface such as a bed boundary.

Two terms used by interpreters, and occasionally throughout this book, need to be defi ned here. First, interpreters sometimes refer to a refl ection produced at the level of a particular horizon as an “event”. For example, if the top of the (hypothetical) Smith Formation generates a single refl ection, it might be referred to as the “Smith event”, much to the chagrin of stratigraphers who use the term “event” to denote something that took place geologically instantaneously (e.g., a volcanic eruption or an extinction event). A geologist recognizes that stratigraphic units such as formations, members, parasequences, etc., are unlikely to have been deposited as a single event (for example, by a single river fl ood). The term is most likely to be used by geophysicists with a broad background in seismology who view seismic traces as amplitude excursions that occur at a particular time (remember, the vertical axis on a seismic trace is time). A second term that may be encountered is “loop”. This term is sometimes used to refer to a single peak or trough between two zero crossings (a positive or negative loop respectively). For example, an interpreter might state that the Smith Formation corresponds to a single positive loop. Both of these terms, event and loop, can be useful when describing seismic refl ections.


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Geometry of Refl ections

Except for some single-channel seismic profi ling systems (such as those used to study modern marine sediments), the energy source and receivers are generally at some distance apart during seismic acquisition. As a result, the energy will hit an interface at an angle, rather than perpendicular to it. If some of the energy is refl ected, then the angle of incidence (Θ1) will be equal to the angle of refl ection as shown in FIGURE 2.24A. Burger et al. (2006) provide a readable account of how basic physics (Huygen’s Principle or Fermat’s Principle) can be used to prove this.

Rays that are transmitted across the interface will be refracted (bent) according to Snell’s Law:

2

1

2

1

sinsin

vv

[2.14]

where Θ2 is the angle of refraction, and v1 and v2 are the velocities of the overlying and underlying medium respectively6. FIGURE 2.25 shows schematically how raypaths will be refracted depending on whether the velocity of the underlying medium is higher (blue), lower (red) or the same (black) as the velocity of the overlying medium. CHAPTER 3 discusses how these types of velocity contrasts can affect seismic imaging.

Note that if the angle of incidence exceeds a certain critical angle (ΘC) the downgoing energy will not be refl ected. Instead, the rays are refracted along the interface and will eventually return up to the surface at the critical angle (FIGURE 2.26). The energy refracted back up to the surface this way is referred to as a head wave, or a fi rst-break refraction. Additionally, in the presence of a velocity gradient, raypaths can bend and eventually move horizontally or even upward without having generated a refl ection. This energy is associated with turning waves, and analyses of these waves is sometimes used to generate velocity profi les or for imaging of steeply dipping features.

Not all refl ecting horizons in the subsurface are planar and horizontal (FIGURE 2.24B) but the angle of incidence will still be equal to the angle of refl ection (as measured with respect to a plane normal to the refl ecting surface), and Snell’s Law will still apply. As such refl ections can take

FIGURE 2.23:

Velocity-density relationships for crystalline igneous rocks and sedimentary rocks

FIGURE 2.24:Acoustic response associated with refl ection of a P wave

FIGURE 2.25:Changes in refraction angle according to whether the velocity of the lower medium is higher, the same, or lower than the overlying medium

__________________________________________________________________

6 Note that this relationship is independent of bulk density.


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unusual paths through areas of dipping strata, especially if velocity contrasts are strong and/or velocity reversals are present. As discussed in CHAPTER 3, dipping interfaces will have an effect on seismic imaging.

FIGURE 2.24 shows the acoustic response associated with refl ection of a P wave. In reality, when a P-wave hits an interface at an angle, some of the energy will be refl ected back as a P-wave and some will be transmitted as a P-wave, as shown in those fi gures. However, because the incident P-wave hits the interface at an angle, some of the wave’s energy will be mode converted into a refl ected S-wave and a transmitted S-wave (FIGURE 2.27). This is known as the elastic response, and we will come back to it when we discuss multi-component seismic data and amplitude variation with offset (AVO) in CHAPTER 8.

Refl ection Amplitude

We now need to discuss how much energy will be refl ected at an interface. For simplicity, we will begin by considering only P-wave energy and vertically incident rays.

P-wave energy will be refl ected at an interface when there is a change in acoustic impedance. The relative amplitude of the refl ection generated at an interface can be predicted using the zero-offset refl ection coeffi cient (R0):

12

120 ZZ

ZZR

[2.15]

where Z1 and Z2 are the acoustic impedances of the layer above and below the interface respectively(FIGURE 2.28). The term “zero offset” effectively means that the sound is hitting the interface at a right angle (i.e. straight down for a horizontal interface). The zero-offset refl ection coeffi cient can be approximated using three other formulae:

*21

0 ZZR [2.15a]

*21

*21

0

R [2.15b]

)ln(21)ln(

21

11220 R [2.15c]

FIGURE 2.26:

Rays refracted along an interface

FIGURE 2.27:Demonstration of the elastic response

FIGURE 2.28:

The zero-off set refl ection coeffi cient


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where ΔZ is the difference in acoustic impedance between the upper and lower layer (i.e. Z2 – Z1)

and Z* is the average acoustic impedance ( 212 ZZ

), Δ is the difference in density between the

upper and lower layer (i.e. 2 – 1) and * is the average acoustic impedance ( 212

), Δα is the difference in P-wave velocity between the upper and lower layer (i.e. α2 – α1) and α* is the average

velocity ( 212

), and ln is the natural logarithm. A signifi cant advantage of EQUATION 2.15B is that it separates the contributions of density and velocity to R0 and this formulation will be helpful in amplitude-variation-with-offset studies (CHAPTER 8).

The refl ection coeffi cient is a relative number, with theoretical limits of -1 to 1, that is unitless. It represents the amplitude of the refl ected wave (Arfl ) divided by the amplitude of the incident wave (Ai)

7:

212

[2.16]

Inspection of EQUATION 2.15 leads to some interesting conclusions. First, if the acoustic impedance of the underlying layer is greater than the acoustic impedance of the overlying layer (Z2 > Z1), R0 is a positive number. In most North American seismic images, a positive refl ection coeffi cient will be displayed as a peak (but see CHAPTER 4 for a discussion of seismic polarity). On the other hand, if the acoustic impedance of the underlying layer is less than the acoustic impedance of the overlying layer (Z2 < Z1), R0 is a negative number which will be represented by a trough in the seismic data. The amplitude of the peaks or troughs, corresponding to the absolute value of R0, can be large or small depending on the relative contrast in acoustic impedance. A large acoustic impedance contrast will generate a strong refl ection, whereas a small acoustic impedance contrast will generate a weak refl ection.

Recall from SECTION 2.3 (FIGURE 2.21) that sands and shales follow different compaction trends in subsiding basins. In the shallow part of the section the acoustic impedance of unconsolidated sand is less than the acoustic impedance of the encasing muds. At depth, the situation is reversed. In the shallow part of such a basin, the top of a sand will correspond to a negative refl ection coeffi cient, whereas deeper in the section it will correspond to a positive refl ection coeffi cient8. Predicting or identifying the depth at which this reversal in acoustic impedance occurs, or whether it occurs at all in the section being studied, can be an important part of a seismic interpretation in thick sedimentary successions (Avseth et al. 2005). This is because an interpreter needs to know how to pick the tops of sand layers (i.e., potential hydrocarbon reservoirs) in the seismic data (i.e. do they correspond to peaks or troughs?).

An important corollary of this discussion is that seismic data image interfaces. In the simplest case (thick bed, zero-phase wavelet) the refl ections correspond to the top (and base) of a bed, and the refl ection amplitude is proportional the magnitude of the acoustic impedance between the layers, not the physical properties of the layers themselves. Of course, predicting the physical properties (porosity, lithology, fl uid content, etc.) of the layers can be the reason for acquiring seismic data

__________________________________________________________________

7 A wave’s amplitude is proportional to the square root of the wave’s energy

8 This discussion assumes that the pore space is entirely fi lled with water.

i

rfl

AA

R 0


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in the fi rst place. A variety of “advanced methods” (see CHAPTER 8) will be needed to convert seismic images of interface properties into images of layer properties. A second consequence of EQUATION 2.15 is that different combinations of layers with different lithologies can produce the same refl ection coeffi cients in which case the amplitude response will be the same. Consider the following two cases:

Lithology α (m/s) ρb (kg/m3) Z (m/s x kg/m3) R0

Case 1 Shale 2450 2300 5635000 0.0346Sandstone 2745 2200 6039000

Case 2 Limestone 4900 2500 12250000 0.0348Dolomite 5150 2550 13132500

Case 1 lists velocity and density values for a relatively young shale and sandstone. If the shale overlies the sandstone, the refl ection coeffi cient generated at the interface will be ~0.035. Case 2 shows values for a limestone overlying a dolomite of higher velocity and density. Note the large absolute difference in velocity and density between the carbonate and clastic example, but the refl ection coeffi cient for both is approximately the same, i.e. the strength of the refl ections from the top of the sandstone and the top of the dolomite will be essentially the same.

For these reasons, seismic data are sometimes referred to as “non-unique”. That is, by itself, the amplitude of a refl ection is not diagnostic of rock properties such as lithology, porosity or type of pore-fi lling fl uids. Based on P-wave refl ection amplitudes alone, we cannot tell whether we are looking at a succession of young clastic sediments or indurated Paleozoic carbonates. However, these can be exactly the types of challenges that confront seismic interpreters. Other lines of evidence (e.g., refl ection confi gurations, variations in amplitude with angle of incidence) will be needed for this purpose, and these topics will be explored in later chapters.

2.5 Convolution and Vertical Resolution

Having described the seismic response for a single interface, it is now necessary to proceed to a more geologically reasonable situation where many layers are present, each with their own velocity and density, and some of the layers are close enough so that the refl ections from the interfaces interfere.

Convolution

Consider the simplifi ed geologic column presented on the left in FIGURE 2.29. Each of the fi ve rock units shown has its own velocity and density, and so its own acoustic impedance. With fi ve layers, we have four interfaces, and it is possible to calculate the refl ection coeffi cient at each interface using the acoustic impedance of the layers above and below each of the interfaces. In the fi gure, the four refl ection coeffi cients are color coded and, in principle, each will generate a refl ection that is a scaled version of the input wavelet. The topmost refl ection coeffi cient (blue) is a strong positive value. This will generate a high-amplitude peak, as shown in blue to the right. Note that the refl ection includes side lobes in addition to the central peak. The next interface down is associated with a small negative refl ection coeffi cient (green) that will generate a low-amplitude


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trough and associated side lobes. The next interface down corresponds to a moderate-amplitude negative refl ection coeffi cient (brown) that generates a moderate amplitude trough. The fi nal interface (red) corresponds to a moderate-amplitude positive refl ection coeffi cient that generates a moderate-amplitude peak and associated side lobes. Unfortunately the individual refl ections are not recorded separately. Instead, the seismic trace recorded at the surface can be considered to consist of the algebraic sum of all the refl ections (four, in this case). This “collective response” is shown to the right.

Mathematically, a better approach than adding the refl ections together is to convolve the wavelet with the series of refl ection coeffi cients. Convolution is a mathematical operation that combines two functions to produce a third function (a multiplication operation for functions). For our purposes, the two functions are: a) the seismic wavelet (our acoustic pulse), and b) the series of refl ection coeffi cients (the geology). We can express this relationship using the following equation:

)()( tgfth [2.17]

where h(t) is the seismic trace, f is the source wavelet, g(t) is the series of refl ection coeffi cients, ε is noise and * is the convolutional operator. The convolutional theorem is a fundamental concept in refl ection seismology which essentially views a seismic trace as the sum of all individual refl ections. It is generally assumed that convolution represents how seismic traces are generated when seismic data are collected in the fi eld.

As an interpreter, it could be our task to look at the trace at the right side of FIGURE 2.29 (a small portion of one of perhaps several hundred or several thousand in a seismic dataset) and defi ne: a) the number of layers, and b) the physical properties of those layers. However there is clearly no simple relationship between the number of peaks or troughs (“events”) and the number of interfaces. The seismic trace includes peaks and troughs generated at the interfaces, but also events generated by side lobes. Additionally, there is no simple relationship between the amplitude of an event and the refl ection coeffi cient. This is because the refl ections (and their side lobes) can either constructively or destructively interfere with overlying or underlying refl ections (and their side lobes). There are

FIGURE 2.29:

Schematic representation of the convolutional theorem

FIGURE 2.30:Seismic modeling experiment to illustrate the importance of wavelet phase on seismic response

FIGURE 2.31:

Schematic representation of a wedge model


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places where amplitude values add constructively (for example a positive refl ection from one level and the positive side lobe from another level) and others where they tend to cancel each other out (e.g., a positive side lobe and a trough). This discussion emphasizes the need to keep side lobes as small as possible, by having a broad-bandwidth wavelet.

In addition to the frequency content of a seismic dataset, the convolutional response is also affected by wavelet phase. Some examples show the importance of wavelet phase on seismic character. The image in the upper left of FIGURE 2.30 shows a simple seismic model with fi ve sand layers that thin and pinch out towards the right side of the model. The other parts of that fi gure show what the geological model would look like if imaged using different wavelets that have the same range of frequencies (5-10-50-70 Hz). Only the zero-phase wavelet has a number of strong peaks that corresponds to the number of sand beds – i.e., the defi nition of stratigraphy is better. The stratigraphy is more poorly defi ned in the minimum phase example because of the prominent reverberations of the input wavelet. The wavelet with the 55° phase also does not properly image the geology. These images illustrate the interpretational advantages of zero phase wavelets. Remember, zero phase wavelets are produced by seismic processing (see section on deconvolution in CHAPTER 3), and cannot be produced with any seismic sources (e.g., dynamite, airguns). Therefore, no clear relationship between the number of peaks (or troughs) and the number of bedding interfaces is to be expected when interpreting seismic records when deconvolution has not been applied (or when deconvolution yields an unstable, reverberating wavelet). This will commonly be the case for shallow (“engineering”) seismic surveys and high-resolution marine profi ling (e.g., boomer sources).

Vertical Resolution, Detection Limits and Tuning Phenomena

The convolutional theorem just described has interesting implications for vertical resolution. Let us imagine a sand bed encased in shale. Both the top and basal interfaces (bed boundaries) will produce a refl ection of different polarity (perhaps a trough at the top and a peak at the base). If the refl ections are widely enough spaced, i.e. the bed is thick enough, each refl ection will be distinct. As the bed thins, the top and base refl ections get closer, and they start to interfere with each other. At a certain bed thickness, the adjacent refl ections become so close that they completely cancel each other out and the bed can no longer be detected seismically. The changes in thickness result in two inter-related phenomena: 1) changes in separation between the refl ections from the top and base of the bed, and 2) changes in amplitude of these refl ections.

Geophysicists investigate these types of relationships using wedge models. For example, imagine a wedge of low acoustic impedance sandstone encased in relatively high acoustic impedance shale such as that shown in FIGURE 2.31A. When the bed is thick enough (e.g., at right), the refl ections from the top and base of the sandstone are separate and the peak/trough separation (in time) is proportional to the bed thickness. Here, the refl ections from the top and base are far enough apart that they do not interfere. As the bed starts to thin, the side lobes begin to interfere with each other. Eventually, the bed thins to a point where the side lobes from the upper refl ection begin to interfere with the main refl ection from the base and vice versa. According to the convolutional theorem (FIGURE 2.31B), this will mean adding negative side lobe amplitudes to the trough at the top, increasing the amplitude of that refl ection, and adding positive sidelobe amplitudes to the peak at the base of the wedge, increasing the amplitude of that refl ection. Although the amplitude of the refl ections changes, the time separation between the two refl ections remains a reliable indicator of


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bed thickness down to λ/4. For a zero-phase wavelet, the side lobes align perfectly with the main refl ections when the bed is λ/4 thick and, because of constructive interference, the amplitude of the refl ections reaches a maximum. This increase in amplitude is known as a tuning effect. Below this thickness, the trough and peak from the top and base of the bed start to interfere destructively and amplitude decreases. Eventually the bed becomes so thin that the refl ections effectively cancel each other out completely. The amplitude of the refl ections decreases. Furthermore, below λ/4 the composite waveform looks like a 90° phase wavelet, similar to a derivative of the input wavelet. The peak/trough separation no longer changes, i.e. it is no longer an indicator of bed thickness.

The λ/4 criterion defi nes the vertical resolution of the seismic data. In this case, the term resolution has a meaning from physics that signifi es that the refl ections from the top and base of the bed can be resolved separately and their separation can be accurately related to bed thickness. Note that it is still possible to detect a bed that is below tuning, even if technically it cannot be resolved. The difference between detection and resolution is a common source of confusion. The detection limit depends on factors, such as signal-to-noise ratio. For example, look at the two wedges shown in FIGURE 2.32. The wedge on the right in FIGURE 2.32A has a higher acoustic impedance contrast with the surrounding medium than the wedge on the left. As such, the amplitudes of the refl ections from the top and base are stronger on the right than on the left. The models shown in FIGURE 2.32A were generated without any noise. FIGURE 2.32B shows the same two wedges, but this time with noise added to simulate poorer data quality. Note that it is now more diffi cult to detect the low-impedance contrast wedge where it is thin. As such, we characterize the signal-to-noise ratio in terms of the signal being related to the strength of the refl ection from a bed (a function of acoustic impedance contrast), and the noise being related to seismic data quality.

FIGURE 2.33 shows amplitude and thickness changes from a wedge model, generated with a “simple” (i.e. few side lobes) 30 Hz Ricker wavelet, rather than a conceptual wedge such as that shown in FIGURE 2.31. The input geological model is shown below, and the seismic modeling result is shown in the middle. Notice how the color bar

FIGURE 2.32:

Effects of noise and acoustic impedance contrast on seismic detectability

FIGURE 2.33:Wedge model showing the effects of bed thickness on seismic response


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emphasizes the increase in amplitudes near λ/4. In reality, we generally do not have perfectly wedge-shaped beds like the one depicted in the fi gure. Instead, this type of modeling is used to illustrate the predicted seismic response for beds of any given thickness.

Sheriff (2002) defi ned resolution as the ability to separate two features that are close together, such as the top and bottom of a bed. Liner (2004) defi ned vertical resolution as the ability to associate peaks (or troughs) on a seismic trace with the top and base of a bed. Kallweit and Wood (1982) discussed various defi nitions of “resolution” and examined the effects of different wavelets, different bandwidths and different types of wedges (e.g., equal but opposite refl ection coeffi cients at the top and base of the wedge, such as shown in Figures 2.31 and 2.33, versus equal refl ection coeffi cients of the same polarity at the top and base) on seismic resolution. They concluded that although there is variability in the predicted seismic response for these cases, the practical limit of resolution is λ/4. Although this resolution is commonly cited in the petroleum industry (e.g., Sheriff and Geldart, 1995), Burger et al. (2006, p. 218) stated that in shallow seismic work (e.g., for engineering purposes) the practical limit is more likely λ/2.

The graph at the top of FIGURE 2.33 shows (in blue) how the isochron (i.e. the time separation) between the refl ection from the top and base of the wedge varies from one end of the wedge to the other. Note that the isochron decreases nearly linearly down to ~ λ/4. Below that thickness, the isochron effectively does not change. Note too how amplitude changes along the wedge. Amplitude does not vary on the right of the graph because there are no interference effects when the bed is thick. As the bed begins to thin, the amplitude starts to increase because of constructive interference between the refl ections from the top and base of the wedge, reaching a maximum value at λ/4 – the tuning thickness. Below tuning, amplitude decreases to zero. Compare these trends with those shown schematically in FIGURE 2.31. Note that the amplitude and isochron trends seen in FIGURE 2.33 are very simplifi ed compared to amplitude and thickness trends that are observed in real data. The reasons for these discrepancies are: 1) Only thickness is changing in the noise-free model, whereas in reality noise, porosity, shaliness and other variables are likely to change along the length of a sandstone bed, and 2) The modeling was done using a Ricker wavelet that has fewer sidelobes than a real seismic wavelet (e.g., FIGURE 2.17).

Castagna et al. (2003) noted that the idea of defi ning a discrete tuning thickness for a seismic dataset may be obsolete. This is because current interpretation technology allows interpreters to view the data at any frequency (within the recorded bandwidth). Because beds of various thickness are likely to be present, each one will have a certain frequency (and associated wavelength, viaEQUATION 2.4) at which the amplitude reaches a maximum because of the tuning effect. Remember that a seismic dataset contains a range of frequencies. By viewing a seismic dataset at different frequencies, it might be possible to see refl ections change amplitude as we change the frequency. We will return to this idea in CHAPTER 7 when discussing an analysis technique called spectral decomposition.

Brown (2004) provided some approximate detection limits (“limit of visibility”) for different scenarios. He suggested that in areas of outstanding signal-to-noise ratio (the signal-to-noise ratio being a function of the strength of the refl ection and data quality), the detection limit might be 1/30 of the wavelength. Detection limits for high, moderate and low signal-to-noise datasets might be


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02

, 21

, and 8

respectively. Sheriff (2002) and Liner (2004) suggested a single detectability limit

of 52

. Note that these are not absolute values, but rather should be thought of as rough estimates

that could be used to make (hopefully) intelligent guesses about detectability limits. Recall that the wavelength can be defi ned using an estimate of the dominant frequency combined with some estimate of velocity (EQUATION 2.4).

Armed with this information, we are in a position to start evaluating questions like: How thick does a bed need to be before it can be seen seismically? Let’s walk through an example. Suppose we know (or suspect) from wireline logs or outcrop analogs that sand beds 2.5 m thick are likely to be present in a certain stratigraphic unit. Should we be able to see them in our seismic data? Perhaps we know from sonic logs that the P-wave velocity of the sands is 3000 m/s, and (from an amplitude spectrum) that the dominant frequency at the target level is 50 Hz. We can prepare the following table:

Velocity Frequency Wavelength Resolution Detection(m/s) (Hz) (m) λ/4 (m) λ/8 (m) λ/20 (m) λ/30 (m)3000 50 60 15 7.5 3 2

If, following Brown (2004), the detection limits for poor, good and outstanding signal-to-noise ratios can be approximated as 1/8, 1/20 and 1/30 of the wavelength respectively, we could predict that we would need good to outstanding data quality in order to see the 2.5 m thick bed. Again, it is important to stress that the numbers listed for the detection limit are approximations designed to help us calculate “ballpark”, not absolute, estimates.

Another utility for this type of analysis would be to determine what types of frequencies would be needed in order to image a specifi c type of target. For example, if we seek to image a 8 - 12 m thick porosity zone in a carbonate reservoir, we might set 10 m as a lower limit for detection. If we know that we can only get “good” (not “excellent”) quality seismic data from an area (perhaps because of surface conditions and/or our budget), we could set 8 m as the lower limit of detection, i.e. λ/20 = 8 m, and the needed wavelength would be 160 m. If the P-wave velocity of the carbonates is 5500 m/s, we can rearrange EQUATION 2.4 to solve for the dominant frequency needed to detect the 8 m thick bed:

DF5500160

We conclude that we would need a dominant frequency of at least 34 Hz in our “good-quality” seismic data in order to have a chance of detecting the porosity zone when it is at its thinnest. This type of knowledge is useful when designing a seismic survey, or when assessing the ability of a particular dataset to image our target.

Before leaving the wedge model, it is important to note that the changes in thickness produce changes in amplitude of the refl ection from the top and base of the wedge shown in FIGURE 2.33. Remember that the physical properties (acoustic impedance) of the wedge and the surrounding shale do not change laterally. These changes in amplitude are known as tuning effects, and they need to


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be understood when using seismic amplitudes to predict physical properties.

Beds less than λ/4 are known as seismic thin beds. Many reservoirs are seismic thin beds. As such, predicting the thickness of these reservoirs (and so, perhaps estimating hydrocarbons in place) from seismic data requires additional information. We noted earlier that the number of refl ections may not equal the number of beds if the wavelet has not been converted to zero phase during processing. However, even if the data are truly zero phase, the number of refl ections may not correspond to the number of beds when working with thin beds, or beds that are so closely spaced that their refl ections interfere with each other. We will return to this point when talking about the stratigraphic interpretation of seismic data in CHAPTER 7.

In reality, it is not just the thickness of the bed that determines whether it will be visible or not. As shown in FIGURE 2.34 it is a combination of the bed’s thickness (with respect to the wavelength) and the degree of acoustic impedance contrast between the bed and the adjacent strata. As labelled at upper right, thick beds that have high acoustic impedance contrasts with the adjacent rocks will be visible seismically. However, relatively thin beds might be visible if the acoustic impedance contrast is high enough, and so might be thick beds with low impedance contrasts. The zone at lower left represents thin beds that have low acoustic impedance contrasts. A bed might have a very large impedance contrast with the adjacent rocks, but if it is too thin it will not be visible (e.g., cm- to dm-scale carbonate concretion layers in shale). A very thick bed might not generate a refl ection if the impedance contrast is essentially zero. Separating the “visible” region from the “not visible” region is a diffuse “grey zone”. Whether or not a bed is visible in this area will depend on factors such as the data quality, and the geoscientist’s interpretation skills.

Changes in Resolution with Depth

One problem that we may encounter during an interpretation is that the frequency content of the seismic dataset commonly changes with depth. The data will typically have more high frequencies in the shallow part of our data than in the deeper parts, especially when working with thick successions represented by several seconds of

FIGURE 2.34:

Schematic graph that predicts whether a bed will generate a detectable refl ection


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data. This problem arises because the farther the sound travels through the earth, the more it loses high frequencies through absorption - the conversion of the wave energy to heat as the waveform passes through the rock and strains it. The energy intensity (I) of a seismic wave at a distance (r) from the source location is related to the source energy intensity (I0) via the following equation:

reII 0 [2.18]

where the absorption coeffi cient η has units of decibels per wavelength. Recall that the higher the frequency the shorter the wavelength (EQUATION 2.4). If the velocity of a rock is 2500 m/s, a 20 Hz wavelet will have a 125 m wavelength and a 50 Hz wavelet will have a 50 m wavelength. To reach a depth of one kilometre and return to the surface, the 20 Hz wavelet will have traveled a distance equal to 16 wavelengths whereas the 50 Hz wavelet will have traveled a distance equal to 40 wavelengths. The higher frequency portion of the signal will therefore lose more energy due to absorption than the lower frequency portion because the absorption coeffi cient in EQUATION 2.18 is in units of decibels per wavelength. The net result is that the farther the sound travels through the earth, the more the high frequencies are lost. This phenomenon is referred to as attenuation.

Another music analog is helpful. Most people have heard the “boom, boom” of music coming through a wall from an adjacent room (e.g., from an annoying neighbor). It is the low notes (low frequencies) that preferentially come through the wall while the high notes (high frequencies) are muffl ed out. The wall acts like a low-pass fi lter, in the same way that the earth preferentially lets the low frequencies pass while fi ltering out the high frequencies.

The absorption coeffi cient η is inversely related to a quantity known as Q, the “quality factor”, which is dimensionless. The quality factor is the ratio of a wave’s energy to the energy dissipated in one cycle of an oscillation. A material that loses no energy has a Q of infi nity, whereas a completely “lossy” material (i.e., one which dissipates all seismic energy) has a Q of zero. Rocks have Q values that range from 50 – 300, with most sedimentary rocks having values near 100 (a seismic wave passing through a rock with a Q of 100 loses 1/100th of its energy every cycle). Grain-to-grain friction causes some absorption in sedimentary rocks, however absorption is primarily caused when motion of the rock matrix and the pore-fi lling fl uids becomes decoupled as the seismic signal propagates through the subsurface (Pride et al., 2003). Much current research is focused on defi ning means of estimating or predicting the quality factor for two primary reasons: 1) it is useful during processing to recover amplitudes or to help during deconvolution (see CHAPTER 3), and 2) it is a physical property that might be useful for estimating permeability or defi ning the presence of hydrocarbons.

We noted previously that velocity tends to increase with depth, and we just noted that the dominant frequency of our seismic data decreases with depth. With depth, the numerator on the right side of EQUATION 2.4 is increasing and the denominator is decreasing. As such, the wavelength increases with depth and the resolution of the seismic data decreases. These effects are shown schematically in FIGURE 2.35. Note that the exact way in which the wavelength increases with depth will vary from basin to basin because of differences in the absorption coeffi cient (absorption tends to be greatest in unconsolidated sediments) and velocity-depth trends. There is effectively nothing that can be done during acquisition to fi x this problem. We might try to use a seismic source rich in high frequencies but, if the earth will not let those frequencies travel down to our target depth and back


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up, we are wasting our time (and probably money). Ideally, we would like to know how the earth will respond during the planning stages of our survey when selecting a seismic source. This may be known if seismic data have already been collected in an area, but it may be an unknown if no seismic data have been collected there.

It should be noted too that time-variant fi lters are sometimes applied to seismic data during processing, with higher frequencies being fi ltered out deeper in the section in an effort to reduce noise. As such, the change in frequency content with depth of a seismic image is controlled both by attenuation and processing.

The wedge model shown in Figures 2.31, 2.32, and 2.33 is one very simplifi ed example of a stratigraphic succession (a bed with sharp basal and upper contacts). The seismic response of other types of stratigraphic successions (e.g., coarsening-upward, fi ning upward) was modeled by Meckel and Nath (1977), Neidel and Poggiagliolmi (1977) and others. FIGURE 2.36 shows how the seismic expression of three simple stratigraphic successions, each consisting of a single layer, changes as a function of thickness. If more than one layer is present, the seismic response will be more complicated because of interference effects (e.g., FIGURE 2.30). In these cases, building seismic models is often helpful for decoding observed seismic responses.

2.6 Lateral Resolution

Until now, we have focused on vertical resolution – how thick something needs to be in order to be detected. We must also think about lateral resolution – how wide an object needs to be in order to be visible seismically.

Lateral resolution is defi ned by the Fresnel Zone. A seismic refl ection is generated from an area, rather than from a single point as might be conceptualized from examination of raypath diagrams. This is because the seismic signal being sent down into the earth forms an expanding wavefront (Figures 2.1, 2.37). The diameter of the Fresnel zone (DF) depends upon three key variables: 1) the average velocity down to the horizon of interest (v), 2) the two-way travel time (t), and 3) the frequency (f):

ftvDF [2.19]

FIGURE 2.35:

Schematic illustration of changes in velocity, dominant frequency of the seismic wavelet and resulting wavelength

FIGURE 2.36:Graphic table showing how changes in bed thickness and abruptness of vertical contacts affects the seismic response for three simple stratigraphic geometries

FIGURE 2.37:Schematic representation of a Fresnel Zone


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The width of an object needs to be equal to or greater than the Fresnel Zone diameter in order to be resolved seismically9. As an example, suppose we are interested in predicting the Fresnel Zone diameter at 2.5 s into the data in an area where the average velocity to that level is 3000 m/s, with a dominant frequency of 25 Hz:

mDF 300525.23000

Although this limitation implies bad news for our ability to detect channels or other features that might be a few 10s of meters wide, in reality the Fresnel zone can be shrunk by a processing step known as migration. Properly migrated seismic data has a Fresnel Zone that is equal to λ/4 (i.e. 30 m for a velocity of 3000 m/s and a dominant frequency of 25 Hz).

2.7 References

Anselmetti, F.S., and G.P. Eberli, 1997, Sonic velocity in carbonate sediments and rocks, in, I. Palaz and K.J. Marfurt, eds., Carbonate Seismology: SEG Geophysical Developments No. 6, p. 53-74.

Avseth, P., T. Mukerji, and G. Mavko, 2005, Quantitative seismic interpretation: Cambridge University Press, 359 p.

Biot, M.A., 1956, Theory of propagation of elastic waves in fl uid-saturated porous solid: Parts I and II: Journal of the Acoustical Society of America, v. 28, p. 168-191.

Brown, A.R., 2004, Interpretation of 3-D seismic data (6th ed.): AAPG Memoir 42, 541 p.

Burger, H.R., A.F. Sheehan, and C.H. Jones, 2006, Introduction to applied geophysics, exploring the shallow subsurface: W.W. Norton and Company, 554 p.

Castagna, J.P., S. Sun, and R.W. Siegfried, 2003, Instantaneous spectral analysis: Detection of low-frequency shadows associated with hydrocarbons: The Leading Edge, v. 22, p. 120-127.

Choquette, P.W., and L.C. Pray, 1970, Geological nomenclature and classifi cation of porosity in sedimentary carbonates: AAPG Bulletin, 54, p. 207-250.

Christensen, N. I., and W. D. Mooney, 1995, Seismic velocity structure and composition of the continental crust: a global view: Journal of Geophysical Research, v 100, p. 9761-9788.

Eberli G. P., G. T. Baechle, F.S. Anselmetti and M.L. Incze, 2003, Factors controlling elastic properties in carbonate sediments and rocks: The Leading Edge, v. 22, p. 654-660.

Fournier, F., and J. Borgomano, 2007, Geological signifi cance of seismic refl ections and imaging of the reservoir architecture in the Malampaya gas fi eld (Philippines): AAPG Bulletin, v. 91, p. 235-258.

__________________________________________________________________

9 Note that the Fresnel zone diameter is defi ned independently of seismic acquisition parameters such as bin size (3-D data) or line spacing (2-D data). Having more closely spaced lines, or smaller bins, does not solve the lateral resolution problem. We return to this issue in CHAPTER 3.


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Gassmann, F., 1951, Ueber die Elastizität poröser Medien: Vierteljahrsschrift der Naturforschenden Ges., v. 96, p. 1-23.

Gardner, G.H.F., L.W. Gardner, and A.R. Gregory, 1974, Formation velocity and density – the diagnostic basics for stratigraphic traps: Geophysics, v. 39, p. 770-780.

Hart, B.S., 2008, Stratigraphically signifi cant attributes: The Leading Edge, v. 27, p. 320-324.

Ji, S., Q. Wang, and B. Xia, 2002, Handbook of seismic properties of minerals, rocks and ores: Polytechnic International Press, Montreal, 630 p.

Kallweit, R. S., and L.C. Wood, 1982, The limits of resolution of zero-phase wavelets: Geophysics, v. 47, 1035-1046.

Liner, C.L., 2004, Elements of 3-D seismology: PennWell, 608 p.

Ludwig, W. J., J. E. Nafe, and C. L. Drake, 1970, Seismic refraction, in, A. E. Maxwell, ed.,The Sea, Vol. 4: Wiley-Interscience, New York, p. 53–84.

Mavko, G., T. Mukerji, and J. Dvorkin, 1998, The rock physics handbook: Tools for seismic analysis in porous media: Cambridge University Press, 329 p.

Meckel, L.D., and A.K. Nath, 1977, Geologic considerations for stratigraphic modeling and interpretation. in C.E. Payton, ed., Seismic stratigraphy—Application to hydrocarbon exploration: AAPG Memoir, 26, p. 417-438.

Milkereit, B., and D. Eaton, 1998, Imaging and interpreting the shallow crystalline crust: Tectonophysics, v. 286, p. 5-18.

Neidell, N.S. and E. Poggiagliolmi, 1977, Stratigraphic modeling and interpretation – geophysical principles and techniques, in, C.E. Payton, ed., Seismic stratigraphy—Application to hydrocarbon exploration: AAPG Memoir, 26, p. 389-416.

Pride, S. R., J. Harris, D.L. Johnson, A. Mateeva, K. Nihei, R.L. Nowack, J. Rector III, H. Spetzler, R. Wu, T. Yamomoto, J. Berryman and M. Fehler, 2003, Permeability dependence of seismic amplitudes: The Leading Edge, v. 22, p. 518-525.

Saleh, A.A., and J.P. Castagna, 2003, Revisiting the Wyllie time average equation in the case of near-spherical pores: Geophysics, v. 69, p. 45-55.

Salisbury, M.H., B. Milkereit, and W. Bleeker, 1996, Seismic imaging of massive sulfi de deposits, 1. Rock Properties: Economic Geology, v. 91, 821-828.

Schön, J.H., 2004, Physical properties of rocks: fundamentals and principles of petrophysics: Elsevier, 583 p.

Sheriff, R.E., 2002, Encyclopedic dictionary of applied geophysics: Society of Exploration Geophysics, Geophysical References Series, 13, 429 p.

Sheriff, R.E., and L.P. Geldart, 1995, Exploration Seismology (2nd Ed.). Cambridge University Press, 592 p.


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Simm, R., and R. White, 2002, Phase, polarity and the interpreter’s wavelet: First Break, v. 20, p. 277-281.

Wang, Z., 1997, Seismic properties of carbonate rocks, in, I. Palaz and K.J. Marfurt, eds., Carbonate seismology: SEG Geophysical Developments No. 6, p. 29-52.

Weger, R.J., G.P. Eberli, G.T. Baechle, J.L. Massaferro, and Y.-F. Sun, 2009, Quantifi cation of pore structure and its effect on sonic velocity and permeability in carbonates: AAPG Bulletin, v. 93, p. 1297-1317.

Wyllie, M.R.J., A.R. Gregory, and G.H.F. Gardner, 1956, Elastic wave velocities in heterogeneous and porous media: Geophysics, v. 21, p. 41–70.


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