operate in an iterative manner, and use some measure of the departure of an individual contribution from the average to down-weight outliers in the next iteration. What is an outlier? As an example, suppose that an attempt is made to estimate the average height of pupils in a school class, but the sample includes four children (1.0, 1.1, 1.1, 1.2 m, respectively) and a basketball player (2.0 m). In this example, the basketball player would be considered to be an outlier. We finally discuss a few ways of displaying the transfer functions, as well as the advantages and disadvantages of various visualisation aids.
4.1 Fourier transformation, calibration and the
(i) Stackingof observations from discretised time windows. (More pre- cisely, the power and cross spectra are stacked in the frequency domain, as we shall see later in this section).
(ii) Weidelt’s dispersion relations (Equation (2.37)) predict that neighbour- ing frequencies will provide similar transfer functions, allowing us to perform a weighted stacking of information from neighbouring frequencies.
We next describe a simple but flexible processing scheme that can be applied either automatically or with user-selected time win- dows. It can be operated in conjunction with aleast-squareestima- tion of the transfer function, or incorporate robust processing or remote reference processingof noisy data (Section 4.2).
Step 1: calculating the Fourier coefficients of one interval of a single time series
Suppose, for example, that a digital time series that represents the Bx-component has been cut into intervals of Ndata points, each of which can be denoted as xj, where j¼1,N (in Figure 4.1, N¼900). Then, by applying thetrend removaland thecosine bell, and thediscrete Fourier transformation,(Equations (A5.12), (A5.13) and (A5.10), explained in Appendix 5), we obtain N/2 pairs of Fourier coefficients am, bm which are combined as complex Fourier coefficients
X~m¼amþibm; (4:1)
where
am¼ 2 N
P
N1 n¼0
xjcosmj
bm¼ 2 N
P
N1 j¼0
xjsinmj 9>
>>
>=
>>
>>
;
m¼1;. . .;N=2:
In the following, we anticipate that all energy above theNyquist frequency,!NY, has been removed from the analogue time series by an anti-alias filter (as described in Section 3.1.4) prior to digitisa- tion. Therefore,
xð!Þ ¼~ 0; 8! > !NY¼ 2p
2t (4:2)
and the discrete Fourier transformation (Equation (A5.10)) recovers the entire information contained in the digital time series xj,j¼1,N.
60 Data processing
In the following discussion, we will denote the mth discrete spectral linesin theraw spectraofBx,ByandBz, andExandEyas X~m,Y~mandZ~m, andN~mandE~m, respectively.
Step 2: calibration
Because a computer can only store integer numbers, a relationship must be established between the output of a sensor and the number stored in a data-acquisition system. This step is calledcalibration5, and consists of four processes.
1 Frequency-independent calibration of the sensors.
We need to know thesensitivity of the sensors. For magnetic sensors, the sensitivity has the unit V T1 or, more usually, mV nT1. For example, the fluxgate magnetometer in Figure 3.2 has a sensitivity SB¼5 mV nT1. For electric amplifiers, the sensitivity has the unit mVðmV=mÞ1 and is calculated from the dimensionless amplifi- cation factor A of the amplifier and the distance, d, between the electrodes:
SE¼Ad (4:3)
2 Frequency-dependentcalibrationof the sensors.
Some sensors, such as theinduction coilin Figure 3.2 have a frequency- dependent sensitivity. In this case, step 1 should be performed at indivi- dual frequencies only. The frequency-dependent sensitivity is described by a dimensionless complex factor wm¼wðfmÞ, where fm is the mth frequency in the output of the Fourier transformation. Any anti-alias filter also has a frequency-dependent complex calibration coefficient, (which is ideally 0 at the Nyquist frequency and 1 at the first evaluation frequency). There are two ways of avoiding the necessity of calibrating anti-alias filters.
(i) In the modern design of instrument described in Figure 3.6 the Nyquist frequency and the first evaluation frequency are so far away from each other that w¼1 at the first evaluation frequency.
(ii) The hardware design of the anti-alias filters of all components is the same and so accurate that estimations of transfer functions (which are ratios of different components of the electromagnetic field) are not affected.
5The termcalibrationis also used to denote estimation of calibration factors related to filters.
4.1 Calibration and the spectral matrix 61
3 Calibration of the A/D converter.
We need to know the resolution of the A/D converter: e.g., a 16-bit A/D converter that can take any voltage in the5V, +5V range has aleast countofl.c.¼10 V=216¼0:153 mV=bit.
4 Application of the calibration coefficient to the output of the Fourier transformation.
In principle, the calibration factors from processes 1 and 3 can be applied to the digital data at any stage in the processing, but obviously the frequency-dependent calibration factorwm from process 2 can only be applied following the Fourier transformation. Some electronic equip- ment (e.g., a steep anti-alias filter) is described by convolution of data with a functionwðfÞ, and this convolution should be performed on the raw spectrum, before the frequency resolution is reduced in step 3:
X~m!l:c:wðfmÞ1S1B X~m; and N~m!l:c:wðfmÞ1S1E N~m:
(4:4)
Step 3: The evaluation frequencies and the spectral matrix
The choice of evaluation frequencies (or periods) is somehow arbi- trary, but two conditions apply:
1 The evaluation frequencies (or periods) should be equally spaced on a logarithmic scale. For example, if we choose 10 s and 15 s as evaluation periods, then we should also choose 100 s and 150 s (rather than 100 s and 105 s), because the relative error dp/pof thepenetration depth(Equation (2.20)) is then the same for periods of the order 10 s and 100 s.
2 Ideally, we should have 6–10 evaluation frequencies per decade – more are unnecessary because Weidelt’s dispersion relation (Equation (2.37)) predicts similar results for neighbouring frequencies, but fewer could result in analiasingin the frequency domain.
A suggested frequency distribution is:
f1¼fmax f2¼fmax= ffiffiffi
2 p f3¼fmax=2
...
fk¼fmax=2pffiffiffiffiffiffiffiðk1Þ :
(4:5)
These evaluation frequencies were used to produce Figure 4.2, withfmax¼0:125 HzorTmin¼8 s:
In Figure 4.1, the sample rate is 2 s and the shortest period is 4 s and therefore,fmax¼0:25 Hz. Practically, however, we want at
62 Data processing
least four samples per period and therefore fmax¼0:125 Hz. The lowest frequency (longest period) is determined by the window length. Typically, we might use a window consisting of 512 data points, but for ease of illustration, the data reduction principle is demonstrated in Figure 4.2 assuming a 128 s time window (i.e, 64 discrete data points), for whichfk¼1=64 Hzandfmax=fk¼8. The raw spectrum from which Figure 4.2 is produced includesM¼32 frequencies, whereas k7 evaluation frequencies result from Equation (4.5). (The first line in the raw spectrum should be ignored, as it is affected by the cosine bell (Equation (A5.13)).
Averaging is performed in the frequency domain. At higher fre- quencies, more information is merged into the same evaluation frequency in order to meet the requirement that the evaluation frequencies are equally spaced on a logarithmic scale. (It could also be stated that, at short periods, we have more independent information per period of the resulting transfer function). The degree of averaging and the form of thespectral windowshown in Figure 4.2 is a compromise between the principle of incorporating as much data as possible (because then the number of degrees of free- dom, which determines the confidence intervals of the transfer functions, will be larger) and not smoothing too much (because mixing different evaluation periods that are associated with differ- ent penetration depths reduces resolution of individual conductivity structures). In other words, the aim is to reach a trade-off between data errors and resolution. In Figure 4.2, a Parzen window (Parzen, 1961; Jenkins and Watts, 1968; Parzen, 1992) is used, but this is only a suggestion.
We next provide an overview of a possible procedure for esti- mating the power and cross spectra (Equations (A6.3) and (A6.4) in
1.0
0.5 gik
k=7
k=1
No 32 No1
∆t=2 s, T0=128 s
f [Hz]
T [s]
mk
0.007125 0.015625 0.03125 0.0625 0.125 0.25 f [Hz]
128 64
2 45.2
3 32
4 22.6
6 16
8 11.2
11 8 16
fNY
Figure 4.2Relationship between the raw spectrum (output of the Fourier transformation) and the evaluation frequencies used for computing MT transfer functions. The form of the spectral window is chosen to optimise the conflicting demands of number of degrees of freedom (which favours including as much data as possible) versus data resolution (which favours not smoothing over too many discrete frequencies). The spectral window weight,gik, is defined by Equation (4.7). Themk
frequencies occurring in the raw spectrum are equi-spaced on a linear scale, whereas the kevaluation frequencies are equi-spaced on a logarithmic scale. Therefore, more smoothing occurs at short periods.
4.1 Calibration and the spectral matrix 63
Appendix 6). For thekth evaluation frequency, we need to know the number of raw data to be merged:
jk¼fk=2f¼Tfk
2 ; (4:6)
wheref is the space between the lines of the raw spectrum and T¼1=fis the span of thetime window. For the Parzen window, theith spectral window weight at thekth evaluation frequency is:
gik¼g0k
sinðip=jkÞ ip=jk
4
; (4:7)
where the scaling constantg0k¼1=jkhas been chosen in such a way that:
g0kþ2Xjk
i¼1
gik¼1: (4:8)
The number of the central frequency of the window is:
mk¼fk=f: (4:9)
The numerical integration depicted in Figure 4.2 is therefore, hX~X~ik¼g0kX~mkX~m
k þXjk
i¼1
gikX~mkiX~m
kiþXjk
i¼1
gikX~mkþiX~m
kþi, (4:10)
whereX~denotes the complex conjugate ofX. Equation (4.10) is a~ numerical estimation of the power spectrum of theBxcomponent at the kth evaluation frequency. Similar equations can be used to compute other power spectra and cross spectra from the data.
These spectra are then stored in aspectral matrixfor each evaluation frequencyk:
X~ Y~ Z~ N~ E~ X~X~ Y~X~ Z~X~ N~X~ E~X~ X~
Y~Y~ Z~Y~ N~Y~ E~Y~ Y~ Z~Z~ Z~ N~N~ E~N~ N~ E~E~ E:~
(4:11)
The number of independent observations or degrees of freedom is k¼2ð1þ2ÞXjk
i¼1
gik=g0k: (4:12)
64 Data processing
If more than one time window is evaluated (which, as mentioned in connection with stacking at the beginning of this section, is normally the case), then the power and cross spectra as well as the number of degrees of freedom from a sequence of time windows are stacked to the previously evaluated data, e.g.,:
hhX~X~ii ¼all windowsX
l¼1
hX~X~il: (4:13)
A proper estimation of a power spectrum (see Appendix 6 for mathematical details thereof ) would require some sort of normal- isation, but for the estimation of transfer functions we are only interested in spectralratios. The stacking procedure can be modified by attributing a weight to the spectra from each particular window.
The weights can be found by hand-editing (in this case they are usually 0 or 1) or with a robust technique, as described in the following section. Recently, Manoj and Nagarajan (2003) applied artificial neural networks in order to find the stacking weights.