2 seismic response

9

Chapter 2

Seismic Response

Seismic response is measured by the reflection generated at an acoustic impedance boundary according to the properties of the layers above and below the boundary and the nature of the seismic pulse impinging on that boundary.

Referring to Figure 1, the equation below defines acoustic impedance (AI) as the product of compressional-wave velocity V and bulk density r:

AI =V

The following equation defines the reflection coefficient (RC) in terms of AI for normal incidence of a seismic pulse at an AI boundary:

RCAI AI

AI AI2 2 1 1

2 2 1 1

2 1

2 1

=( )+( ) =

( )+(

V V

V V

)) .

The Zoeppritz equations define the reflection coefficient for nonnormal angles of incidence of a seismic pulse at an AI boundary; these equations generally are applied in a simplified form (e.g., Shuey, 1985). For the pur-poses of this text and defining seismic as having to do with elastic waves (Sheriff, 2002), here we describe seismic response in terms of compres-sional-wave (P-wave) reflections but do not discuss shear waves (S-waves) or mode conversions in detail.

You can initially and most easily describe seismic response with refer-ence to an isolated impedance boundary and can further develop understand-ing of the composite response from multiple, closely spaced boundaries by way of the convolutional model (discussed later in this chapter). You need

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10 First Steps in Seismic Interpretation

to be familiar with a mathematical description for a waveform in terms of its frequency, amplitude, and phase characteristics, being especially careful to define phase and polarity as used in describing the shape or character of a reflection. The confidence with which you identify and correlate a reflec-tion from an acoustic impedance boundary, which interpreters call a seismic event or horizon, based on its appearance or character depends on seismic data quality, on simple and well-known impedance relationships, and, per-haps most importantly, on correlation of seismic data to available well data via well ties.

The importance of horizon identification increases as you move along the value stream from wildcat exploration through appraisal and devel-opment to production because this movement is toward greater detail of description in telling your geologic story. When interpreting and mapping in a frontier area, it may not be important to know whether a particular reflec-tion corresponds to the top of a sand or a shale. But for a production project in the same area many years and millions of dollars later, it could be crucial to understand the seismic response for the top of a reservoir sand when choosing well locations and calculating reserves hence, the importance of understanding seismic response in identifying horizons for interpretation.

Understanding the seismic response to an AI boundary requires knowl-edge of the seismic pulse incident to that boundary and the behavior of the

Figure 1. Definitions of acoustic impedance (AI) as a rock property, defined as the product of compressional-wave velocity V and bulk density r. The contrast in AI between two layers of rock gives rise to a seismic reflection when a seismic pulse impinges on the boundary between the layers.

V = compressional-wave velocity, r = bulk density

V1, r1

V2, r2

Upper layer

Lower layer

Incidentpulse

Reflectedpulse



Chapter 2: Seismic Response 11

pulse as it propagates through the earth. The seismic pulse causes particle motion in the subsurface through a medium treated as elastic in response to stress applied in the form of an impulse (e.g., detonating a charge of dynamite or firing an air gun). Dix (1952, his Figures 11.4 and 11.5) presents schematic diagrams illustrating these particle motions for positive and negative reflec-tion processes. A seismic waveform is a description of this particle motion as a function of time, which can be treated as a composite of many individual functions of time for the different frequency components present in the wave-form; the analytical representation of a seismic waveform as the sum of indi-vidual sinusoidal functions is called Fourier analysis (Sheriff, 2002).

For the sake of clarity and proper use of terminology, you should always be careful to distinguish between a reflector and a reflection: the former is a surface or boundary across which there is an acoustic impedance contrast, and the latter is a measurement of the particle motion caused by impinge-ment of a seismic pulse upon the former. Keep in mind that you observe reflections and interpret reflectors (that is, elements of geology) from your observations in that order. Maintaining a clear distinction between reflec-tions and reflectors will help you remember that no seismic line or volume, no matter how carefully acquired and processed, is a completely accurate representation of true subsurface geology.

A seismic pulse propagates through a subsurface that is not really elas-tic, so you cant expect the pulse to retain its exact shape as it travels from the seismic source to a receiver. The change in shape of a wavelet, which is to say in the amplitude and phase characteristics of its different frequency components, because of propagation through a nonelastic earth is called attenuation. The physical properties of the subsurface of the earth cause the higher-frequency components of a wavelet to be preferentially reduced in strength, primarily because of converting the energy of particle motion to the heat of friction. In general, the farther or longer a signal travels, the more it is attenuated. Attenuation correction of seismic data, which can be done probabilistically (based on measurements of the data themselves) or deterministically (based on correlation with other physical measurements) is an important step in a seismic data-processing sequence.

The change in shape of a wavelet as a result of attenuation suggests that, all other things being equal, you should not expect to see the same seis-mic response to the same impedance boundary that occurs at two different depths. A modeled product such as a synthetic seismogram, which usually is generated with an invariant wavelet, will therefore be better for making an accurate well tie in that portion of the seismic section where the wavelet used for the synthetic seismogram is a good approximation for the actual wavelet in the data. This is why wavelets are extracted from seismic data




over windows or intervals of specific interest and then are used to generate synthetic seismograms for correlation only in that interval. Where possible, these extractions are done at or near points of well control so that log data can be used in the extraction process.

In the time domain, a periodic function for a single frequency can be described as a sinusoidal wave, as with the cosine wave illustrated in Fig-ure 2. The general form of the equation for this cosine wave as a function of time is

y t A ft( ) cos ( ),= +2

where A is the amplitude, f the frequency, t the traveltime, and the phase of the waveform. The value is the angle, measured in degrees (where 360 = 1 cycle), that represents the lead (the amount of time the waveform is advanced) or lag (the amount of time the waveform is delayed) with respect to a reference starting time. Phase is defined as the negative of phase lag (Yilmaz, 2001), which is to say that a negative time shift (time delay) cor-responds to a positive phase value and a positive time shift (time advance) corresponds to a negative phase value. For example, Figure 3 shows that a cosine wave lags a sine wave by /2 or 90:

sin cos cos( ) , sin cos 2 2 2

0 1 0

=

= = ( ) = 00

2 20

=

=

cos ,. . .

or

cos sin sin( ) , cos sin 2 2 2

0 0

= +

= = ( ) = 00

2 21+

=

=

sin ,. . . .

Figure 2. A simple sinusoid defined as a cosine wave. The shape of this waveform is determined by its amplitude A, frequency f, and phase . T is the period of the waveform.

t

T

A




The waveforms shown in Figures 2 and 3 are infinite, single-frequency sinusoids; however, all of the wavelets with which you work in practical seismic interpretation are finite and have limited bandwidth. They are the summation of discrete sinusoids, each with its own amplitude, frequency, and phase characteristics. This is the basis of Fourier analysis. An exam-ple of a finite, band-limited wavelet and its component sinusoids is shown in Figure 4; in this example, the amplitude and phase of the components are constant (phase = 0) and only the frequency of the individual sinusoids varies.

Knowledge of the phase of a waveform is important in Fourier analysis because this angle sets a reference for the starting time (zero time, effec-tively) for each component waveform defined by its own frequency and amplitude. An illustration of phase rotation of a simple band-limited wave-let symmetric about t = 0 through one full cycle from 0 to 360 for 90 increments is shown in Figure 5. As expected, phase rotations of 180 and 180 are identical.

The wavelet in the center trace in Figure 5 is symmetric about t = 0, meaning that it literally describes particle motion that occurs before t = 0, which is physically nonrealizable. For this reason, the wavelet is called a non-causal wavelet (see Figure 6). Because of its symmetry, it is also referred to as a zero-phase wavelet; each of its component sinusoids is zero phase, and each is uniquely defined by its own amplitude and frequency according to Figure 2. In terms of signal processing, a zero-phase wavelet has the shortest time duration (pulse width) for a given bandwidth (frequency range). The

Figure 3. Phase relationship between a sine wave (red) and a cosine wave (blue). The sine wave leads the cosine wave by 90, and the cosine wave lags the sine wave by 90.

cos(0) = sin(0 + p /2) = sin(p /2) = 1sin(0) = cos(0 p /2) = cos(p /2) = 0

t

p /2 p /2 3p /2 2pp0




Figure 4. Illustration of a finite, band-limited wavelet as the summation of five component sinusoids. All of the components have the same amplitude and phase (phase = 0).

Finiteband-limited

wavelet

40 Hz 30 Hz 20 Hz 10 Hz 5 Hz

t

Figure 5. Phase rotation of a zero-phase wavelet (center trace) through a full 360 in increments of 90. The display convention used in this figure is described in Figure 7.

0180 90 +90 +180

Tim

e

+

_




seismic response for a zero-phase wavelet also is easier and more intuitive to visualize because its maximum amplitude corresponds exactly to the posi-tion of the reflecting interface (see Figures 5 and 6). Displays that show the amplitude and phase characteristics of the sinusoids for every frequency component of a wavelet are called the amplitude (amplitude as a function of frequency) and phase (phase as a function of frequency) spectra. Given these amplitude and phase spectra, a resultant wavelet can be uniquely con-structed by summing individual frequency components having the charac-teristics defined by these spectra.

Figures 5 and 6 use the same display convention, i.e., they represent seismic response in the same way with reference to a standard impedance configuration. The display convention most commonly used by SEG is the positive standard polarity convention (Figure 7), in which polarity means positive or negative trace deflection. When discussing or presenting your work, you should state the phase of your data, to the degree it is known, and the display convention you are observing. Similarly, you should ask about wavelet phase and the display convention being used in any discussion or presentation involving seismic data if that information is not communicated or clearly annotated on seismic displays.

Figure 8 illustrates the four different display formats for reflection seis-mic data. Of these, the most common used on workstation displays is vari-able density, often with user-defined or customized color schemes. Wiggle traces superimposed on a variable density background is also a popular dis-play format.

Figure 6. Noncausal and causal wavelets. The causal wavelet involves particle motion only after time = 0, whereas the noncausal wavelet involves particle motion before time = 0, which is not physically realizable. The display convention used in this figure is described in Figure 7.

0

Causalwavelet

Noncausalwavelet

Tim

e

+

_




In virtually all cases, reflection seismic data represent a composite response to many closely spaced impedance boundaries, some of which are sharp and distinct and others of which are gradational. This composite response actually is the result of constructive and destructive interference of the discrete responses to individual impedance boundaries, described by the so-called convolutional model. Convolution is a mathematical operation that, in simplest terms, involves multiplication, shifting, and summation of two functions of the same variable (for seismic data the variable is traveltime t). You can think of convolution as simulating the propagation of a seismic pulse through a layered earth. The output of a 1D convolution, such as the convolution of an RC series calculated from an AI log (which has been converted to the time domain) with a seismic wavelet to produce a synthetic seismogram is probably much easier to visualize than to describe in words or to understand from exacting math-ematical language.

In Figure 9, the RC series consists of four coefficients, each correspond-ing to an AI boundary; the coefficients are not evenly spaced, and they do not all have the same magnitude and sign. This RC series will be convolved with the zero-phase wavelet shown to the left of the series, and both must have the same sample rate. Note that this wavelet is a wiggle trace that uses the SEG positive standard polarity convention. In the convolutional model, the seismic response to a given RC is created by reproducing the seismic wavelet scaled to the magnitude and sign of that RC. As shown in Figure 9, the scaled wavelet is reproduced as the seismic response for each of the four RCs, and the final convolution output or composite response is

Figure 7. The SEG positive standard display convention for reflection seismic data. For a zero-phase wavelet, a positive reflection coefficient is represented by a central peak, normally plotted black on a variable area or variable density display (Sheriff, 2002).

Acoustic impedance

Reflection coefficient

+ Low

High

Wavelet

0




Figure 8. Four display formats for reflection seismic data. Display formats are independent of the polarity convention used for a given data set.

Variable density Variable area Variable-area wiggleWiggle

the sum of the individual scaled responses. There is both constructive and destructive interference between individual seismic responses in the com-posite response. This interference is substantial when the effective width of the seismic pulse is greater than the interval between adjacent RCs. For purposes of this discussion, consider the pulse width to be the breadth of the central peak or peak/trough. Notice also that there is no individual seismic response for any points in the RC series where RC = 0, that is, where there is no impedance contrast. The differences between the composite responses in Figure 9a and 9b indicate that your interpretation of geology from seismic data depends critically on the wavelet in your data.

Knowledge of wavelet phase is important because it relates seismic response to geology in terms of the characteristics of the source wavelet (pulse) as defined in Figure 2, that is, the reflection seismic response to a given geologic boundary or feature changes for different source wavelets. The phase of the wavelet contained in any seismic data set can vary laterally and vertically (temporally) and is estimated most accurately by determin-istic methods using well control. In the absence of well control, you can




Figure 9. (a) The convolutional model. The individual responses of each reflection coefficient to the input seismic wavelet, scaled to the magnitude and sign of the reflection coefficient, are summed to generate the composite seismic response. There are destructive and constructive interference of the individual responses in producing the composite response. (b) Convolution of the reflection coefficient series shown in (a) with a different source wavelet. The differences between the composite responses for the two wavelets show that accurate interpretation of these responses depends on knowledge of the source wavelets.

Input wavelet

Reflection a)coefficients

Individual responses

Overlay responses

Composite response

+

Input wavelet

Reflection b)coefficients

Individual responses

Overlay responses

Composite response

+

visually estimate wavelet phase by observing certain reflections that may be present in your data (see Table 1).

Using reflections from any of the boundaries listed in Table 1 assumes that the boundary can be identified conclusively, that there is a well-known and consistent acoustic impedance contrast across it (the algebraic sign of the reflection coefficient across the boundary is known), and that it is isolated




from other nearby boundaries so that its character is not a composite reflec-tion response. In marine settings, the seafloor reflection is commonly used to check wavelet phase because the impedance contrast between seawater and sediment is almost always positive. Similarly, a hydrocarbon/water contact, which appears as a seismic flat spot in a reservoir that is thick enough to be resolved seismically, can be used confidently to estimate wavelet phase (see the discussion of seismic resolution and tuning in Chapter 6). A seismic flat spot occurs because the presence of hydrocarbons as the pore-filling fluid lowers the AI of the hydrocarbon-bearing portion of a reservoir below that of the nonhydrocarbon-bearing or brine-filled portion of that reservoir. Not all flat spots are perfectly flat because velocity effects in time imaging can tilt or distort them and because some hydrocarbon/water contacts are not truly horizontal. A flat spot can occur only for reservoirs in which the hydrocarbon-bearing portion of the reservoir is seismically resolved because the seismic response from a hydrocarbon-bearing interval whose thickness is below a cer-tain value called the tuning thickness will be a composite of responses from the top and base of the interval that will not directly represent wavelet phase.

The flat spot indicated by the arrow in Figure 10 shows a well-defined, symmetric peak (black). According to the accepted polarity standard and display convention for this image, within the visual acuity of the observer to see asymmetry in the waveform, the phase of the data is zero. Note that near the right-hand edge of this flat spot is a high-amplitude trough-over-peak amplitude response; this point marks the tuning thickness of the low-imped-ance, hydrocarbon-bearing portion of the reservoir. Continuing to the right, the decrease in the amplitude of the trough-over-peak signature reflects the decrease in thickness of the hydrocarbon-bearing portion of the reservoir. Note also that the top of the reservoir is not marked by a single, sharply defined reflection (a trough or a peak) along its full extent, suggesting that the top of the reservoir interval might be gradational in some places.

Table 1. Subsurface boundaries that can be used for visual estimation of wavelet phase. No single boundary is absolute or foolproof.

Best:

Seafloor

Hydrocarbon/water contact (seismic flat spot)

Use with care:

Top of salt/volcanics

Base of salt

Basement




The problem with using boundaries such as top and/or base of salt, top of volcanics, and basement (which can take on a variety of geologic and eco-nomic meanings) for estimating wavelet phase is that these boundaries often are gradational and poorly defined, so their seismic responses are effec-tively composite responses to multiple, closely spaced impedance contrasts rather than to a single, well-known impedance contrast. At the same time, the impedance properties of the materials above and below these boundar-ies, especially for basement, are not necessarily well known or regionally consistent; so neither the magnitude nor the sign of the impedance contrast across such boundaries can be inferred confidently without well control.

Most interpreters prefer to work with zero-phase data, for which a seis-mic event or horizon is symmetrically disposed about its correlative imped-ance boundary and thus is most easily and intuitively visualized. Knowledge of wavelet phase and the display convention of your data should enable you to draw geologically reasonable conclusions when correlating a given seis-mic response to a particular AI boundary. At the same time, you should rec-ognize that a given impedance boundary can give rise to different seismic responses, depending on the phase of your data. This knowledge is critical for accurate interpretation of seismic attributes, as discussed in the next chapter.

Figure 10. Example of a well-imaged seismic flat spot, denoted by the yellow arrow, on time-migrated data. This image suggests that the seismic data are zero phase (courtesy PGS).

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