Sampling and aliasing

As described in Section 2.2, a geophysical measurement may consist of one reading or a series of readings made at the same location. Measurements made over a period of time or range of frequencies form time and spectral series, respectively, for that location. Whether a single reading or a series of readings is made at each location, a survey will consist of measurements at a number of stations to form a

spatial series. These series constitute a set of samples of what is a continuous variation in the parameter being measured. In order for the samples to accurately represent the true variation, these series must be appropriately sampled, i.e. readings must be taken at an appropriate spacing or interval.

The rate at which the sampling occurs is known as the sampling frequency, which for time series is measured in units of 1/time (frequency, in units of hertz (Hz)); for frequency series, 1/frequency (period, in units of seconds (s)); and for spatial series, 1/distance (units of reciprocal metres (m-1)). The time, frequency or distance between samples is the sampling interval. A little confusingly, spatial sampling for a moving sensor is usually defined in terms of temporal sampling. For example, detectors used in aeromagnetic surveys measure (or sample) the Earth’s magneticfield typically every 0.1 s, i.e. 10 times per second, referred to as 10 Hz sampling. Afixed-wing aircraft acquir- ing magnetic data flies at about 70 m/s, so at ’10 Hz sampling’ the spatial sampling interval is 7 m.

If the measurements are not spaced closely enough (in time, frequency or distance) to properly sample the par- ameter being measured, a phenomenon known as aliasing occurs. It is respectively called temporal, spectral or spatial aliasing. Consider the situation where a time-varying signal being measured varies sinusoidally. The effect of sampling at different sampling frequencies to produce a time series is demonstrated inFig. 2.8a. The true variations in the signal are properly represented only when the sampling fre- quency is high enough to represent those variations; if not, the signal is under-sampled or aliased. When under- sampling occurs, the frequency of the sine wave is incor- rectly represented by being transformed to spurious longer wavelength (lower frequency) variations. Clearly then, the very act of sampling a signal can produce artefacts in the sampled data series that are indistinguishable from legit- imate responses. Strategies can be adopted to combat the problem of aliasing, but if data are aliased it is impossible to reconstruct the original waveform from them.

2.6.1.1 Sampling interval

It is shown inAppendix 2that a complex waveform can be represented as a series of superimposed sine waves each with a different frequency. To avoid aliasing a waveform containing a range of frequencies, it is necessary to sample the waveform at a sampling frequency greater than twice the highest frequency component of the waveform. This is the Nyquist criterion of sampling (Fig. 2.8b). In

spatial terms, this means that the interval between meas- urements must be less than half the wavelength of the shortest wavelength (highest frequency) component. Con- versely, the maximum component frequency of the signal that can be accurately defined, known as the Nyquist frequency (fN), is equal to half the sampling frequency

(fs). The interval between zero frequency and the Nyquist

frequency is known as the Nyquist interval. Frequency variations in the input waveform occurring in the Nyquist interval are properly represented in the sampled data series, but frequencies higher than the Nyquist frequency

are under-sampled and converted (aliased) to spurious lower-frequency signals, i.e. they are ‘folded back’ into the Nyquist interval. The aliased responses mix with the responses of interest and the two are indistinguishable. The aliased responses are artefacts, being purely a product of the interaction between the sampling scheme and the waveform being sampled.

The sampling interval required to avoid aliasing can be established with computer modelling (see Section 2.11), reconnaissance surveys or field tests conducted prior to the actual survey. In practice, economic and logistical considerations mean that aliased data are often, and unavoidably, acquired. This is not necessarily a problem for qualitative interpretations such as geological mapping or target detection. Regions with similar characteristics will usually retain apparently similar appearance even if alias- ing has occurred, but different geology will give rise to different geophysical responses. Extreme caution is required when the data are to be quantitatively analysed, i.e. modelled (seeSection 2.11.3), because working with an aliased dataset will result in an erroneous interpretation. 2.6.1.2 Example of aliasing in geophysical data

Figure 2.9shows an example of spatial aliasing in magnetic

data collected along a traverse across the Elura Zn–Pb–Ag volcanogenic (pyrrhotite-rich) massive sulphide deposit located in New South Wales, Australia. Data collected at a station spacing of 25 m (minimum properly represented wavelength 50 m) show variations with wavelengths of

0 200 400 600 800 1000 Location (m) Mineralisation Mineralisation 250 nT 0.25 m Station spacing 25 m Station spacing 0.25 m Station spacing (filtered)

Figure 2.9Aliasing in total magnetic intensity data across the Elura Zn–Pb–Ag massive sulphide deposit. See text for details. Redrawn

from Smith and Pridmore (1989), with permission of J. M. Stanley,

formerly Director, Geophysical Research Institute, University of New England, Australia.

100 0 Output frequency (Hz) a) b) 50 Sampling frequency > 2f_In Sampling frequency = f_In Input frequency (fIn) (Hz) Sampling frequency < fIn

Sample Input waveform Waveform after sampling Time Under-sampled Nyquist interval Nyquist frequency (fN) Sampling frequency (fS) 400 0 50 100 150 200 250 300 350

Figure 2.8 Aliasing of a periodic signal of frequency fIn. (a) Signal

and various sampling frequencies; see text for explanation. (b) The relationship between the frequency of the input signal and the frequency of the sampled output signal for a sampling frequency of,

say, 200 Hz. Note how under-sampling causes the output to‘fold-

back’ into the Nyquist interval (0–100 Hz), e.g. an input signal of 250 Hz produces a sampled output signal of 50 Hz. Redrawn, with

50 m and more, a consequence of the Nyquist criteria. Importantly, there is no noticeable difference in the data acquired above the mineralisation and elsewhere. The near-surface material contains occurrences of the magnetic mineral maghaemite, which produce very short-wavelength (high spatial frequency) variations (seeSection 3.9.6). Data acquired at a smaller station spacing of 0.25 m (minimum properly represented wavelength 0.5 m) are more useful as they show not only the longer wavelength variation of the mineralisation, but also the very short-wavelength‘spikey’ variations of the near-surface. The near-surface variations are under-sampled in the 25-m-sampled dataset and, there- fore, have created spurious longer wavelength variations; the true signature of the near-surface response has not been resolved by the survey. The short-wavelength near-surface response is properly defined in the 0.25-m-dataset and, therefore, can be accurately removed using data processing techniques (see Section 2.7.4), to reveal the longer wave- length response of the mineralisation (Fig. 2.9). This example illustrates the need to consider the characteristics of both the signal and the noise when setting the data sampling interval, and the implications that the sampling interval has for the ability to separate the various responses using data processing techniques.