Data Acquisition & Reduction

Geophysical Survey Geophysical surveys measure variation of some physical quantity, with respect to either to position or to time. Space: Magnetic field variation which may be measured along a profile. Time: Seismogram showing variation of particle velocities in the ground as a function if time

Data Acquisition and Processing The first stage of geophysical methods is making measurements in the field (data acquisition). Then further stages working with the data (data reduction), before that are ready for geological deductions to be drawn (data interpretation)

Data Acquisition and Processing Data acquisition and reduction - describes the necessary basic stages, from taking measurements to converting the data into a relevant form. It also includes graphical ways of displaying the results more clearly. Data processing - describes special mathematical ways for separating wanted (signal) from unwanted (noise) parts of the results.

Data Acquisition: Taking Measurements Most geophysical measurements are made at the Earth’s surface. Often the instrument readings are taken along a line or traverse. Reading are not taken continuously along the traverse but are taken at intervals. Each place where a reading is taken is called a station. When the readings are plotted they form a profile.

Data Acquisition: Taking Measurements cont… Often, several parallel profiles are taken to see how the body continues. If the traverses are close to one another the stations form an array or grid, and the results are often contoured. Converting the results into a more useful form.

Signal Classification Signals can be broadly divided into two main classes Continuous signals Are function of a continuous variable (e.g. continuous or analogue signal) Discrete signals Defined only at particular values (e.g. daily measurements of water level) Consist of evenly spaced samples The interval between samples, T, is called the sampling interval (ms, or m) Sampling frequency, 1/T, is the number samples recorded for each unit.

Digitization of geophysical data Waveforms are generally continuous (analogue) functions of time or distance. Computers requires the data to be expressed in digital form. A continuous, smooth function of time or distance can be expressed digitally by sampling the function at a fixed interval.

Digitization of geophysical data cont… Digitization depends on: Accuracy of the amplitude measurement (sampling accuracy or dynamic range). Intervals between measured samples (sampling frequency)

Dynamic Range Expression of the ratio of the largest measurable amplitude Amax to the smallest Amin. The higher the dynamic range, the more faithfully the amplitude variation will be represented in the digitised version. Expressed in the decibel (dB) scale. Dynamic range is given by: 20 log10 (A1/A2) Example: Amplitude range is 1 to 1024 units of amplitude . Therefore, the dynamic range is: 20 log10 1024 ~ 60 dB

Dynamic Range cont… In computers digital samples are expressed in binary form (digits that have the value of either 0 or 1). Each binary digit is known as a bit, and sequences of bits is known as a word. Number of bits in each word determines the dynamic range of a digitized waveform. Dynamic range of 84 dB represents an amplitude ratio of 214, and requires 15-bit words. A dynamic range of 48 dB requires 9-bit words since the appropriate amplitude ratio of 256 (=28) is rendered as 100000000 Increasing the number of bits in each word increases the dynamic range.

If a digital recording of a geophysical time series is required to have a dynamic range of 60 dB, what number of bits is required in each binary word? 9 10 11 12 None of the above :120

Sampling frequency Is the number of sampling points in unit time or unit distance. No data is lost as long as the frequency of sampling is much higher than the highest frequency component. A drawback of sampling the signal is that wavelengths that do not exist can appear to be present.

Data acquisition and reduction Dynamic range and sampling frequency What are the factors that determine the digitization (converting the signal from continuous to discrete) of a signal? Data acquisition and reduction Dynamic range and sampling frequency Space and Time Data processing and interpretation :30

Aliasing The loss of high frequency information from a continuous signal during sampling. In the case of a sampling frequency of 500 Hz Input frequency of 125 Hz is retained in the output Input frequency of 625 Hz is also folded back to the output at 125 Hz.

Aliasing cont… To overcome the problem of aliasing, the sampling frequency must be at least twice as high as the highest frequency (fn). E.g. to record seismic signals in the frequency range from 0 to 50 Hz, the minimum sample frequency must be 100Hz If the function contains frequencies above fn, it must be passed through an antialias filter prior to digitization.

The Nyquist Frequency What is the highest frequency signal which can be recorded for a given sampling interval? For a discrete signal with a sampling interval T the highest frequency which can be restored is fn=1/2T This is called Nyquist Frequency Frequency of half the sampling frequency E.g. for a sample interval of 2ms the Nyquist frequency is 250 Hz Frequency above this is aliased or “Folded back” to frequencies < 250 Hz

Alias Frequency The apparent frequency of a signal due to inadequate sampling Aliasing occur if frequencies above fn presented in the sampled data. Can be calculated from relationship: fa=2fn-fs Where fn is the Nyquist frequency, fs is the signal frequency E.g. a 65 Hz signal sampled with a sampling interval of 8 ms (fn=1/2T)=62.5 Hz The alias frequency is fa=2x62.5-65 = 60Hz

Nyquist frequency Signal frequency Antialias filter Sampling frequency What is the highest frequency signal which can be restored accurately for a given sampling interval? Sampling frequency Alias frequency Nyquist frequency Signal frequency Antialias filter :60

10 Hz 50 Hz 125 Hz 250 Hz None of the above Time-series data are sampled at 4 ms intervals for digital recording. What Nyquist frequency should be used to prevent aliasing? 10 Hz 50 Hz 125 Hz 250 Hz None of the above :120

10 Hz 25 Hz 50 Hz 125 Hz None of the above In the absence of antialias filtering, at what frequency would noise at 225 Hz be aliased back into the Nyquist interval? 10 Hz 25 Hz 50 Hz 125 Hz None of the above :120

Summary Continuous signals are typically sampled at discrete sample intervals. The sampling interval must be short enough to allow adequate interpolation of the discrete samples. The sampling frequency must be greater than twice the highest frequency component in the original signal. For a sampling interval, T, the highest frequency which can be accurately restored (Nyquist frequency) is fn=1/2T High frequency signals which are inadequately sampled are said to be aliased.

Data Processing Getting More Information from the Data

Review We discussed the concepts of signals and sampling: Continuous signals are typically sampled at discrete sample intervals. The sampling interval must be short enough to allow adequate interpolation of the discrete samples. The sampling frequency must be greater than twice the highest frequency component in the original signal. For a sampling interval T the highest frequency which can be accurately restored (Nyquist frequency) is fn=1/2T High frequency signals which are inadequately sampled are said to be aliased.

Data Processing: Getting More Information from the Data Even after the results of a survey have been reduced and displayed, the features of interest may not be obvious. If so, there may be further stages of processing that will enhance the features.

Signal and Noise A waveform is a combination of signal and noise. Signal is the part of the waveform that relates to the geological structures under investigation. Noise is all other components of the waveform Random noise: due to effects unconnected with the geophysical survey. e.g. background noise due to wind, rain, or traffic. Coherent noise: components of waveform which are generated by the geophysical experiment, but are of no direct interest. e.g. surface waves generated by the seismic source, multiples.

Signal and Noise cont… The geophysicist’s task is to separate the signal from the noise and interpret the signal in terms of ground structure. In seismology, noise can be vibrations due to passing traffic or anything that shakes the ground.

Signal and Noise cont… Noise can also be spatial. In magnetic survey noise may be due to wire fences or buried bits of metals, which can obscure the signal. What is a noise – and what is signal – may depend on the purpose of the survey

Signal and Noise cont… One way to improve the signal-to-noise ratio is to repeat readings and take their average (stacking). The signal parts of each reading add, where the noise, usually being random, tends to cancel A more general method to make the wanted signal more clearer is to use signal processing. Filters enhance the signal by reducing unwanted wavelengths, such as the short wavelengths due to noise.

The purpose of stacking is to: Display the result more clearly Improve the signal-to-noise ratio. Help reduce the data. None of the above All of the above :30

Wavelength and Amplitude This curve is called sinusoid, because it is described by the mathematical sine function. The repeat distance (measured between successive crests) is the wavelength, . The maximum deviation from the undisturbed position is the amplitude, a.

Periodic and Transient Waveforms Periodic waveforms repeat themselves at a fixed time period T. Transient waveforms are non-repetitive.

Fourier Analysis Periodic waveform, can be decomposed into a series of sine (or cosine) waves. Fourier Analysis is a mathematical technique that sorts features by their width (wavelength) or frequency, from which we can then select the ones we want.

Fourier Analysis: Time & Frequency Domain A periodic waveform can be expressed either in time domain or in the frequency domain The time domain is a graph of the dependent variable (e.g. voltage) as a function of time. In the frequency domain a signal is characterised by variation of both amplitude and phase as a function of frequency

Fourier Analysis: Time & Frequency Domain Time and frequency domain representation of a waveform, g(t) and G(f), are known as a Fourier pair g(t)  G(f) Fourier transform pairs for: A spike function A DC bias Seismic pulses

Frequency Filtering Involves transformation of the time domain signal into the frequency domain and modification of the amplitude prior to conversion back into the time domain. Multiplication of the Fourier transform of the signal by a filter function which has a value of 1 for frequencies which will be unaffected, decreasing to zero for those frequencies which will be attenuated.

Frequency Filtering cont… Frequency filtering involves altering the frequency content of a signal to enhance a selected component. Low pass (LP) filters preserve the low frequency components and attenuate the high frequency. High pass (HP) filters preserve the high frequency components and attenuate the low frequency. Band pass (BP) filters preserve intermediate frequencies while attenuating both high and low frequencies. Notch filters attenuate a narrow frequency range in the signal without affecting other frequencies.

Frequency Filters cont… They are employed when signal and noise components of a waveform have different frequency characteristics. Complete Bouguer Anomaly RegionalBouguer Anomaly Residual Bouguer Anomaly

Fourier Analysis: Example – Component of a gravity anomaly A granite was intruded below an area in the past and later was exposed by erosion. The uneven surface of the granite and surrounding country rock is buried beneath overburden.

Fourier Analysis: Example – Component of a gravity anomaly Wavelength analysis can be used to sort the features by their wavelength. The profile due to the granite alone consists mostly of long wavelength, while the overburden profile has shorter wavelengths.

Fourier Analysis: Example – Component of a gravity anomaly The unwanted anomalies of the overburden are an example of noise. As noise has a short wavelength than the signal, removing shorter wavelengths improves the signal-to-noise ratio. Short wavelength anomalies are due to near surface bodies Broad anomalies are due to either a narrow body at depth or to a broad body near surface.

Which of the following is true Wide anomalies are always due to deep bodies Narrow anomalies are always due to shallow bodies The deeper a body, the wider its anomaly None of the above All the above :30

Correlation and Convolution Are important mathematical processes which are of fundamental importance in signal processing. Correlation is used to measure the similarity between two signals. Convolution is used to calculate the effect of a physical system on a signal. Both processes can be implemented in either the time or frequency domain.

Convolution A mathematical operation defining the change of shape of a waveform resulting from its passage through a filter. The effect of a filter is described mathematically convolution operation. y(t) = g(t)  f(t) Where the asterisk denotes the convolution operator y(t) is the filtered output g(t) is the input signal f(t) is the impulse response The impulse response of a filter

Convolution cont… The form of a filtered output is the summation of a set of impulse responses

Convolution cont… The convolution of the source wavelet (1,0,2) and the reflectivity sequence (4,3,2,1) is calculated by: Aligning the moving array with the stationary array and multiplying corresponding elements Adding the products calculated Shifting the moving array a lag of one to the right and repeating above steps for all possible lags.

Convolution cont… The convolution of the source wavelet (1,-0.5) and the reflectivity sequence (1,0,0.5) is calculated by: Reflectivity Output 1 0 0.5 -0.5 1 1 -0.5 1 -0.5 -0.5 1 0.5 -0.5 1 -0.25 The output is the time series (1, -0.5, 0.5, 0.25)

If the digital signal (1, 3, -1, 1) is convolved with the filter operator (2, 1, 3), what is the convolved output? 1,3,-1,1 2,1,3 3,1,2 1,-1,3,1 None of the above :200

Convolution – Frequency Domain Convolution of two function in the time domain is laborious as the function become longer. Same results can be obtained by transforming the functions to the frequency domain Multiplying together equivalent frequency terms of their amplitude spectra Adding terms of their phase spectra. The resulting output amplitude and phase spectra can be transformed back to the time domain.

Convolution (1)

Convolution (2) Impulse response R(t) Output O(t) Iutput I(t) Process A Output O(t) Iutput I(t) Process O(t) = I(t)*R(t)

Convolution (3) Impulse response R(t) Output O(t) Iutput I(t) Process A Output O(t) Iutput I(t) Process t O(t) = I(t)*R(t)

Multiplication Addition Division None of the above Convolution of two functions in the time domain is equivalent to what in frequency domain (frequency terms)? Multiplication Addition Division None of the above :30

Correlation Many signal processing applications including seismic processing require the measurement of the similarity or time alignment of two signals. To understand the process of correlation, consider the two signal wavelets: A(t) = (2,1,-1,0,0) and B(t)=(0,0,2,1,-1) It is clear that the form of the signal is the same for A(t) and B(t) The signals would be the identical if B(t) was shifted backwards by two time steps.

Correlation cont… The correlation is calculated by: Cross-multiplication of the individual waveform elements Summation of the cross-multiplication products. Correlation in time domain is equivalent to multiplication of amplitude spectra in the frequency domain Subtraction of phase spectra in the frequency domain.

Correlation cont… The correlation of two different wavelet is called cross-correlation Cross-correlation function measures the degree of similarity of waveforms. Useful in the detection of weak signals embedded in noise.

Correlation cont… The correlation of a signal with itself is called autocorrelation which is always a symmetrical zero phase wavelet. Is useful in the detection and suppression of multiple reflection in seismic records.

Modelling Model is a body or structure (described by such properties as depth, size, density, etc.) that could account for the data measured. Modelling Types Inverse Modelling Forward Modelling

Displaying the Results Profiles Contouring Isometric projection

Summary In time domain the signal is characterised by the variation of a signal variable (amplitude) as a function of time. In frequency domain, the same information is defined by variations in two variables (amplitude and phase) as a function of frequency. A signal in time domain comprises a wide range of frequencies. Variations in phase effect the symmetry and location of the signal pulse in the time domain. Signal conversion between time and frequency domain is accomplished using Fourier Transform

Summary Correlation compares two signals and provides a numerical measure of their similarity. Convolution calculates the effects of a signal on a system calculated in either the time or frequency domain. Convolution is used to calculate the response when a signal is passed through a filter. Convolution of two signals in the time domain is equivalent to multiplication of the transforms of the signals in the frequency domain .