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APPENDIX 2 AND ANALYSIS

A2.1 Introduction wave terminology and, importantly, they can be manipu- lated and analysed as if they were waves recorded in the The term wave is used to describe a localised disturbance space domain. which travels through some medium. Consider a cork are frequently encountered in geophysics, so floating in water. A ripple, i.e. a wave, travelling (propagat- it is essential to be familiar with the terminology used for ing) across the water’s surface causes the cork to bob up and describing them and the mathematical methods used for down. The movement of the cork (and of individual water manipulating them. Specific aspects of the various type of molecules) is due to the disturbance associated with the waves used in geophysical surveys, and their propagation wave. The wave can be thought of as a moving packet of in the case of moving waves, are described for each strain energy travelling through the water. Although the geophysical method in the relevant chapters of this text. water is deformed (strained) during the passage of the ripple, Here we provide only a general description of waveforms it only produces an oscillation about a point. The cork does and wave-related phenomena and the techniques used in not move laterally, demonstrating that the disturbance their analysis. associated with the wave does not represent a flow of water. In the example above, the variation in the surface height of the water could be plotted as a function of location for A2.2 Parameters defining waves different times (Fig. A2.1). Each of these plots is a spatial and waveforms series (see Section 2.2). The movement of the ripple A familiar and extremely useful is a wave, through space is clearly apparent when the four spatial also referred to as a sinusoid. This is a graph of the sine series are compared. It is also possible to plot the height function from trigonometry, where the sine of an angle is of the water surface at some chosen point as a function of plotted against the angle. As will be demonstrated in time (Fig. A2.1b). This is a (see Section 2.2) and Section A2.4, other waveforms can be represented as the in the example represents the cork bobbing up and down sum of a series of different sine waves. as the ripple passes. A is continuous. A waveform is said to be A graph of the variation of some physical parameter continuous if it goes to infinity in either time or distance. related to a wave is known as a waveform. As shown in The concept of a is fundamental to the Fig. A2.1 this can be plotted in the time domain or the space definition of a wide variety of waveforms. There also exists domain depending on the type of recording. Although these another class of waves which are said to be discontinuous. represent variations in two different physical dimensions, These have limited temporal/spatial extent, i.e. they only mathematically they can be treated identically so they can exist in a restricted area of time or space, and are known be manipulated and analysed in the same way. The seismic as wavelets. An example of a wavelet in the time domain is a and electromagnetic waves exploited in geophysical surveys single hand clap: the only exists in a restricted inter- travel through the subsurface and are usually analysed as val of time/space – it is a non-continuous wave. The equiva- time series recorded at a stationary recording location. lent continuous sound wave might be a continuous hum. However, graphs of the spatial variations of any of the Referring to Fig. A2.2a, a number of parameters define physical quantities measured in geophysical surveys, a wave: although not necessarily directly created by wave phenom- ena, have all the attributes of a waveform (see Fig. 1.1a in • The maximum variation in the sine wave, which the main text). They can be conveniently described using accounts for the positive and negative variations, is 2 Waves and wave analysis

a) a) Space domain Magnitude + Direction of + propagation difference (Df) (A)

Location To–¥ To¥ A t Time/location – Time = 0

+ Direction of #2 B #1 propagation (l) – or period (P ) Time/space domain Location Magnitude

t Dt b) – Time = 0+ ) A )

+ Direction of #1 Amplitude f Phase propagation difference #2 #2 difference #1 (Df) Phase ( Location Amplitude (

Magnitude Frequency C t Dt – Time = 0+2 Direction of + propagation D Figure A2.2 Continuous sine waves. The representation of two continuous sine waves in (a) the time/space and (b) the frequency domains. See text for further explanations. Location Magnitude Magnitude

– Time = t +3 Dt • A sine wave repeats itself. The repeated section of the 0 Wavelength (l) Reference wave is known as a cycle. This is the section between two point consecutive equivalent points on the waveform experi- encing identical disturbance. For example, the section b) Magnitude (at reference point) between two adjacent peaks or troughs, or two zero – + cross-overs of the same slope etc., represents one cycle A t of the wave. Time = 0 B Time • In the space domain, the distance that one cycle of the Time = t + Dt 0 domain sine wave occupies is known as the wavelength (λ)of C Time = t +2D t Period (P ) 0 the wave. The time domain equivalent of wavelength is t D t the period (P) of the wave, which is the time taken for Time = 0+3 D one cycle of the wave to occur. Time • In the time domain, the number of repetitions or cycles per unit of time, i.e. cycles per second or (Hz), is Figure A2.1 Waveforms. (a) Waveforms showing the magnitude of a the frequency (f) of the wave. In the space domain, the wave’s disturbance in the region of a reference point in a medium at different times (separated by time interval Δt) as a wavelet passes number of cycles per unit of distance, e.g. cycles per through the point, and (b) the disturbance at the reference point metre, is the or wave number (σ)of shown as a waveform varying in time. the wave. It is common for spatial frequency to be (incorrectly) referred to as frequency. • fi known as the peak value or amplitude (A) of the wave. In The various parameters de ned above are related as the water-ripple analogy, this is the height of the ripples shown in the following Eqs. (A2.1) to (A2.4): above or below the average water level. As described in σ ¼ 1 ð : Þ Section 6.3.3.1, the amplitude of a wave is a related to its λ A2 1 energy content. The term amplitude is also used in a less 1 formal to represent the magnitude of the waveform f ¼ ðA2:2Þ at some specific time or location. P A2.3 3

For a wave travelling at a velocity (V) a) Magnitude

isc d g T = 2 T V n = 1 ¼ ð : Þ ti f A2 3 a t λ o T = 3 R T = 0 and so T = 4 Axis 0 1234 Time (T ) V λ ¼ ðA2:4Þ f Df • A sine wave of a particular frequency can also be b) Magnitude Phase difference described in terms of a radial line of a reference circle isc d T g = 1 n T = 0 ti rotating anticlockwise, i.e a rotating disc (Fig. A2.3a). The a t o T = 2 length of the line is the amplitude (A) of the wave R f

(Fig. A2. 2a). The variation in the vertical height of the T = 3 Axis 0 1234 Time (T ) end of the amplitude line, measured above and below T the axis of the disc, as it rotates is the magnitude of the = 4 waveform with time. Df • The rotating disc analogy demonstrates clearly the con- Phase c) Magnitude cept of phase angle (ϕ), often just called the phase. In the difference (180°) isc d three parts of Fig. A2.3 the disc starts rotating with the g n ti a amplitude line in different positions. The three wave- t o f = 180 forms are shifted relative to each other, parallel to the R T = 4 T = 0 0 1234 Axis Time (T ) time axis. The amount of shift depends on the degree T = 3 T = 2 of rotation of the disc at the different starting positions. T = 1 This is why phase is an angle. • Figure A2.2a shows two sine waves of the same frequency and with different . The offset Figure A2.3 Rotating disc analogy for the magnitude variations of a between them is their phase difference (Δϕ) (Fig. sine wave. See text for details. A2.2b). As shown, the variation represented by wave- form #1 leads that represented by waveform #2 by the amount of their phase difference. Alternatively, we can amplitude as a function of frequency and known as the say that waveform #2 lags waveform #1 by the phase amplitude spectrum, and the other showing the phase difference. as a function of frequency and known as the phase • An important case is when the phase difference between spectrum. The frequency domain representation is called two waveforms is 180° (π ), i.e. they are com- the frequency spectrum, Fourier spectrum, or simply the pletely out of phase. This is the case for the waveforms in spectrum. Figs. A2.3a and c. There is an exact alignment of the peaks of one waveform with the troughs of the other A2.3 Wave interference waveform, which is equivalent to multiplying the ampli- tude of one waveform by –1, i.e. reversing the wave- Waves interact with each other through a process known form’s polarity. When the waveforms are perfectly in as interference. It is an important property of waves which step, i.e. their peaks (and their troughs and cross-overs) occurs where two or more waves occupy the same space. are coincident, their phase difference is zero and the These combine to produce a resultant wave whose proper- waves are in phase. ties depend on the relative amplitudes, and phases of the individual waves. The magnitude of the An alternative to the time/spatial domain representations resultant at any location is the sum of the magnitudes of of a sine wave is its frequency domain (also called Fourier the individual waves at that location. Where the peaks, and domain) representation. In Fig. A2.2b, the two sine waves troughs, of the waves tend to coincide the magnitude of the are represented by two graphs: one showing variations in resulting disturbance is greater than that of the individual 4 Waves and wave analysis

waves. This is known as constructive interference. Where spectrum tells us what frequencies are present in the wave- the peaks tend to coincide with the troughs the resultant form and in what proportions they occur. In other words, magnitude is less and the effect is known as destructive it displays the waveform’s frequency content and tells us interference. how much of the variability in the waveform is due to low- A waveform of any type can be described in terms of a frequency waves (long ) and how much is group of interfering sine waves of different frequencies, related to high-frequency waves (short wavelengths). Also, and possibly with different amplitudes and phases. It is the frequency domain representation is a mathematically very convenient to describe and characterise a waveform in more convenient and efficient way of manipulating and terms of its constituent sine waves, i.e. in terms of its analysing waveforms, particularly complex waveforms (see frequency spectrum (see Section A2.2). The amplitude Section A2.4).

Time/space domain Frequency/Fourier domain

Resultant Waveform #1 Waveform #2 Resultant a) waveform waveform

Waveform #1 Amplitude Amplitude Amplitude Interference Interference (S) (S)

Phase Phase Phase

Waveform #2 Frequency Frequency Frequency

b)

Waveform #1 Amplitude Amplitude Amplitude Interference Interference (S) (S)

Phase Phase Phase

Frequency Frequency Frequency

Waveform #2 (change in amplitude) c)

Waveform #1 Amplitude Amplitude Amplitude Interference Interference (S) (S) Phase Phase Phase

Waveform #2 (change in phase) Frequency Frequency Frequency d)

Waveform #1 Amplitude Amplitude Amplitude Interference Interference (S) (S) Phase Phase Phase

Waveform #2 (change in frequency) Frequency Frequency Frequency

Figure A2.4 The interference of two continuous sine waves illustrating how differences in their amplitudes, phases and frequencies determines the nature of the resultant wave, which is shown in both the time/space and frequency domains. (a) Waveform #2 is twice the frequency of waveform #1 and their amplitudes are equal; (b) the amplitude of waveform #2 has increased; (c) the amplitudes of both waveforms are the same as (a) but the phase of waveform #2 has changed; and (d) the amplitude of both waveforms is the same as (a) but the frequency of waveform #2 has increased. A2.4 Spectral analysis 5 a) Figure A2.4 shows the effects of varying the amplitude, 1 phase and frequency of just one of two interfering waves. Notice how the occurrences of constructive and destructive interference change position along the resultant waveform, 5 radically affecting its shape and its frequency spectrum. The concept of a geophysical dataset comprising a series of interfering waveforms is of fundamental importance. In

10 the main text there are numerous references to, and dis- cussions of, longer- and shorter-wavelength (lower- and higher-frequency) components of the variation in the data. The waveform in Fig. A2.4d might be described in these 15 terms. Often these represent signal or (see Section 2.4). For example, the problem of separating shorter- 20 wavelength (residual) variations from longer-wavelength b) (regional) variations is discussed at length in Section 2.9.2.

Sum of waveforms 1 to 5 Dominant period A2.4 Spectral analysis Sum of waveforms 1 to 10 A waveform can be transformed into its frequency spec- trum, or separated into its component sine waves, through Sum of waveforms 1 to 15 a process known as spectral analysis. The mathematical operation used for the spectral, or Fourier, analysis of Sum of waveforms 1 to 20 waveforms is called the . A forward transform converts a time or spatial series into its Fourier To –¥ To ¥ 0 domain equivalent. As the name suggests, the reverse or Time/location c) inverse transform does the opposite. The Fourier trans- Amplitude Phase Sum of waveforms 1 to 5 form is a complex algorithm the mathematics of which are 5 beyond our scope, but a description containing minimal mathematics is provided by Rayner (1971) and a more 1 1 5 0 mathematical, but still easy to follow description, is pro- Frequency Frequency vided by Dobrin and Savit (1988).

Amplitude Phase When transformed into the frequency domain, a Sum of waveforms 1 to 10 10 5 waveform can be modified in ways to suit the user, i.e. the amplitudes and or phases of some or all of the com- 1 1 5 10 0 ponent sine waves can be changed, for example those with Frequency Frequency

Figure A2.5 Amplitude Phase Summing zero-phase wavelets. (a) A series of sine waves Sum of waveforms 1 to 15 10 15 of different frequencies and amplitudes whose positive peaks are 5 coincident in time/space at the central point (because they all have 1 1 5 10 15 zero phase shift). (b) Various zero-phase wavelets shown in the time/ 0 Frequency Frequency spatial domain obtained by summing different combinations of the sine waves shown in (a). The black horizontal line represents the

Amplitude Phase dominant period. Note how the period decreases, and the pulse Sum of waveforms 1 to 20 10 15 5 narrows, as more high-frequency sine waves are included. Vertical scale varies. (c) The amplitude- and phase-spectra of the wavelet 1 20 1 5 10 15 20 created by summing the sinusoids in (a). Phase is zero across the 0 Frequency Frequency bandwidth because all the components are zero-phase sinusoids. 6 Waves and wave analysis

a) particular frequencies removed. The process of modifying fi f the original waveform in this way is called ltering, and 1 0 is commonly utilised when the signal and noise have different frequency content; see Section 2.7.4.4. 3f 2 0 Figure A2.4 demonstrates that quite complex waveforms 5f 3 0 can be created from the interference of just two sine waves. f 4 7 0 f Even complicated waveforms that bear little resemblance 5 9 0 f 6 11 0 to a sine wave can be created from the summation of 13f 7 0 different sine waves. Figures A2.5 and A2.6 demonstrate 15f 8 0 this for two waveforms of relevance to geophysics. The wavelet in Fig. A2.5 might represent the pulse of energy b) created during a seismic survey (see Chapter 6) or a high- frequency electromagnetic () survey (see Appendix 5). Sum of waveforms 1 to 2 The in Fig. A2.6 is a waveform commonly created by the used in electrical and electro-

Sum of waveforms 1 to 3 magnetic surveys (see Chapter 5). Referring to Fig. A2.5, each of the sine waves added to create the wavelet has a different frequency, most have the Sum of waveforms 1 to 4 same amplitudes and in this case all have zero phase. This makes the sinusoids symmetrical about a central point

Sum of waveforms 1 to 5 where they all have coincident peaks. The wavelet is also symmetrical about the central point and this is where its peak value occurs. The wavelet is referred to as a zero- Sum of waveforms 1 to 6 phase wavelet. As more and more sine waves of higher and higher frequency are added, the wavelet becomes shorter. It is said to be more localised in time (or space) but it has a Sum of waveforms 1 to 7 larger range in its frequency content (spectrum), i.e. it has greater bandwidth. A zero-phase wavelet is the wavelet that

Sum of waveforms 1 to 8 has the shortest time/spatial duration for a given frequency content. It is common also to talk about a dominant frequency or dominant period of a wavelet, which is that

To –¥ To ¥ of the central peak. 0 Time/Location The limiting case is when the wavelet is so short it is c) everywhere zero except at one time/location where it has Amplitude Phase finite amplitude. This is referred to as a spike or impulse. f 0 A perfect impulse is formed by summing sinusoids of fi 0 equal amplitude across all frequencies, i.e. it has an in nite 3f f 0 f f f 5 0 f 0 0 0 f f f f 7 9 0 f f f f 0 0 0 11 f 0 0 0 0 0 13 15 3 5 7 9 11 13 15 bandwidth. Frequency Frequency The square wave is created by the summation of sine

Figure A2.6 Square waves. (a) A series of sine waves with waves whose frequencies are odd multiples (n) of the frequency increasing as the odd multiple (n) of the fundamental lowest or (f0), whose amplitudes

frequency (f0) and amplitudes decreasing as 1/n as shown and decrease with increasing frequency by the inverse of the described in the text. (b) Various approximations of a square wave multiple (1/n), and they all have zero phase (Fig. A2.6). As shown in the time/spatial domain obtained by summing different more high-frequency sine waves are added to the series the combinations of the sine waves shown in (a). Vertical scale varies. ‘teeth’ of the square wave become better-defined, i.e. their (c) The amplitude- and phase-spectra of the square wave created fl by summing all eight sine waves. Phase is zero across the spectrum edges are stepper and their tops atter. The perfect square because all the components are zero-phase sinusoids. wave requires an infinite number (n goes to infinity) of sine waves to be summed together. References 7

The frequency content of a square wave makes it a very ‘sampled’ to be processed using computers. The spacing useful waveform when measurements are required at between samples needs to be smaller when the data contain different frequencies. It is commonly exploited to great variations of higher frequency. In practice there are limits to advantage in electrical geophysics where multi-frequency how small the sample interval can be. This means the entire measurements are made simultaneously. The recording information content in the data is not represented by the can be reduced to its component sine waves using the sampling and so the Fourier-transformed data are not an Fourier transform, and the amplitudes and phases of exact representation of the actual data. If there are discontinu- selected frequencies analysed. ities in the data, i.e. abrupt changes such as steps or spikes, The ‘sharpening’ of the square wave and localisation of there is a tendency for spurious short-wavelength reverber- a wavelet with the addition of high-frequency sine waves ations to appear in the transformed data. The effect is known has important implications for geophysical data analysis. as ringing.ItismoreproperlyknownasGibbs phenomenon,a As described in Section 2.6.1, geophysical responses detailed description of which is beyond our scope. This is a comprise continuous waveforms that have to be digitally form of methodological noise (see Section 2.4.2).

REFERENCES

Dobrin, M.B. and Savit, C.H., 1988. Introduction to Geophysical Rayner, J.N., 1971. An Introduction to Spectral Analysis. Pion. Prospecting. McGraw Hill.