Least square, remote reference and robust estimation of transfer functions

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~i_l: (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.

4.2 Least square, remote reference and robust

Least-square and robust processing techniques are examples of statistical processing methods commonly employed to remove noise from MT data. In both techniques, Equation (4.15) is solved as a bivariate linear regressionproblem (see Appendix 4). Least-square processing techniques involve isolating the components ofZusing least-square cross-power spectral density estimates, with the func- tionZð Þ! assumed to have aGaussian (normal) distribution. Cross- powers for a discrete frequency can be generated in the frequency domain by multiplying Equations (4.14a) and (4.14b) by the com- plex conjugates (denoted^*) of the electric and magnetic spectra as follows:

hExð ÞE! _xð Þi ¼! Z_xxð ÞhH! xð ÞE! _xð Þi þ! Z_xyð ÞhH! yð ÞE! _xð Þi! (4:16a) hExð ÞE! _yð Þi ¼! Z_xxð ÞhH! xð ÞE! _yð Þi þ! Z_xyð ÞhH! yð ÞE! _yð Þi! (4:16b) hE_xð ÞH! _xð Þi ¼! Z_xxð ÞhH! _xð ÞH! _xð Þi þ! Z_xyð ÞhH! _yð ÞH! _xð Þi! (4:16c) hE_xð ÞH! _yð Þi ¼! Z_xxð ÞhH! _xð ÞH! _yð Þi þ! Z_xyð ÞhH! _yð ÞH! _yð Þi! (4:16d)

hEyð ÞE! _xð Þi ¼! Zyxð ÞhH! xð ÞE! _xð Þi þ! Zyyð ÞhH! yð ÞE! _xð Þi! (4:16e) hEyð ÞE! _yð Þi ¼! Z_yxð ÞhH! xð ÞE! _yð Þi þ! Z_yyð ÞhH! yð ÞE! _yð Þi! (4:16f)

hEyð ÞH! _xð Þi ¼! Z_yxð ÞhH! xð ÞH! _xð Þi þ! Z_yyð ÞhH! yð ÞH! _xð Þi! (4:16g) hEyð ÞH! _yð Þi ¼! Z_yxð ÞhH! xð ÞH! _yð Þi þ! Z_yyð ÞhH! yð ÞH! _yð Þi:! (4:16h) Most of these Equations (e.g., Equation (4.16a)) contain auto- powers. Since any component will be coherent with itself, any noise present in that component will be amplified in the auto-power, causingZijto be biased. One solution to this problem is theremote reference method.

The remote reference method (Goubau et al., 1979; Gamble et al., 1979; Clarkeet al., 1983) involves deploying additional sen- sors (usually magnetic) at a site removed (remote) from the main (local) measurement site. Whereas the uncontaminated (natural) part of the induced field can be expected to be coherent over spatial scales of many kilometres, noise is generally random and incoher- ent. Therefore, by measuring selected electromagnetic components at both local and remote sites, bias effects arising from the presence of noise that is uncorrelated between sites can be removed.

Correlated noise that is present in both local and remote sites cannot be removed by this method. The distance between local and remote

66 Data processing

sites that is necessary in order to realise the assumption of uncorrel- ated noise depends on the noise source, intended frequency range of measurements and conductivity of the sounding medium. In the past, the separation distance between local and remote sites was often determined by equipment limitations, owing to the require- ment that local and remote sites should be linked by cables.

However, with the advent of GPS clocks, which allow instruments to be precisely synchronised, simultaneous data can be ensured without the need for cable links, and local and remote instruments can be placed many kilometres apart. Magnetic remote reference is favoured over electric, because horizontal magnetic fields exhibit greater homogeneity than electric fields in the vicinity of lateral heterogeneities, are less susceptible to being polarised, and are generally less contaminated by noise than electric fields. Using the notation from Section 4.1 and adopting subscripts r to denote the remote reference magnetic fields, Equations (4.16) can be solved for Z_ijgiving:

Z_xx¼hN~X~_rihY~Y~_ri hN~Y~_rihY~X~_ri DET

Z_xy¼hN~Y~_rihX~X~_ri hN~X~_rihX~Y~_ri DET

Z_yx¼hE~X~_rihY~Y~_ri hE~Y~_rihY~X~_ri DET

Z_yy¼hE~Y~_rihX~X~_ri hE~X~_rihX~Y~_ri

DET ; ð4:17Þ

where

DET¼ hX~X~_rihY~Y~_ri hX~Y~_rihY~X~_ri:

Now, in order to calculateZat a particular frequency, we simply substitute the appropriate power spectra and cross spectra, which we computed in Section 4.1 and stored in the spectral matrix described by Equation (4.11), into Equations (4.17). The impedance tensor calculated from Equation (4.17) will not be biased by noise provided that the noise in the local measurements is uncorrelated with the noise recorded by the remote reference configuration, but a high degree of correlation between the naturally-induced electromagnetic fields at the local and remote sites is required. In multi-dimensional environments noise may be exaggerated in the cross-spectral density terms, as well as in the auto-power terms.

Recently, Pa´duaet al.(2002) showed that remote reference methods may be no better than single-station processing techniques at removing the kind of noise generated by DC railways.

4.2 Estimation of transfer functions 67

A more sophisticated use of a remote reference site involves a two-source processing technique, in which the electric field expan- sion (Equation (4.15)) includes a second magnetic field (which represents the noise in the local magnetic field) and a second imped- ance (which represents the noisy part of the electric fields that is correlated with the magnetic noise). By using the remote reference site to separate these two sources of noise, it is possible under certain circumstances to remove noise which is correlated between the mag- netic and the electric readings at the local site (Larsenet al., 1996).

An instructive field example is given by Oettingeret al., (2001).

The elements of the induction arrow,T(!)(Equation (2.49)), are estimated similarly toZ(!)(i.e., by multiplying by the complex conjugates of the horizontal magnetic fields to give):

T_x¼hZ~X~ihY~Y~i hZ~Y~ihY~X~i DET

Ty¼hZ~Y~ihX~X~i hZ~X~ihX~Y~i DET

9>

>>

=

>>

>;

: (4:18)

MT transfer functions should vary smoothly with frequency (Weidelt, 1972). Therefore, if large changes in MT transfer func- tions are apparent from one frequency to the next it may be inferred that they have been inaccurately estimated.

We calculate the confidence intervals of the transfer function as detailed in Appendix 4 (Equation (A4.20)) with a modification that since twocomplextransfer functions have been estimated, the term 2=ð2Þis replaced by4=ð4Þ. For example,

T_x

j j²g_F4;4ðÞ4"²hZ~Z~ihY~Y~i ð4ÞDET T_y

²g_F4;4ðÞ4"²hZ~Z~ihX~X~i

ð4ÞDET ;

(4:19)

where the residuum is estimated according to Equation (A4.17):

"²¼1 ²¼1ThZ~X~iþThZ~Y~i hZ~Z~i

: (4:20)

Can you prove that the bivariate coherency ²is real?

Fundamental to least-square statistical estimation is the assump- tion of power independent, uncorrelated, identically distributed Gaussian errors, whereas Egbert and Booker (1986) have shown that deviations from a Gaussian error distribution occur owing to error magnitudes being proportional to signal power, and owing to

68 Data processing

episodic failure of the uniform source field assumption (particularly during magnetic storms). The clustering of some related errors (for example, during magnetic storms) further undermines the assump- tion of uncorrelated errors. Other techniques for eliminating bias arising from data points (often denotedoutliers; see analogy shown in Figure 4.3) that are unrepresentative of the data as a whole include so-called robust processing techniques (e.g., Egbert and Booker, 1986). Robust processing involves a least-squares type algorithm, but one that is iteratively weighted to account for departures from aGaussian distributionof errors. By comparing the expected versus observed distribution of residuals, a data-adaptive scheme, reliant on a function (e.g., regressionM-estimate, Huber, 1981) containing iteratively calculated weights is implemented, with data points poorly fitting the expected robust distribution being assigned smaller weights (Egbert and Booker, 1986):

X^2N

i¼1

w_ir²_i: (4:21)

The weights, w_i, are inversely proportional to the variance (1=²_i) of the ith data point from the mean andri is the residual.

The residual distribution is Gaussian at the centre, but is truncated and, therefore, has thicker (Laplacian) tails. If data points assigned weights below a pre-determined threshold are omitted from the stacking process, this results in a reduction of the number of degrees

ITERATION 1

Heights (cm) of children in a school class Number of children 0

2 4 6 8 10

20 40 60 80 100 120 140

RAW DATA MEAN

Heights (cm) of children in a school class Number of children 0

2 4 6 8 10

20 40 60 80 100 120 140

WEIGHTED MEAN Number of degrees of freedom = 23 0

1

Number of degrees of freedom = 31

σ

}

OUTLIERS wi

wi

∝

= 1^{1 σσii< σ>σ} σi2

σOUTi

σOUTi

Figure 4.3Trivial example of how robust processing can be used to downweight outliers.

Notice that downweighting is performed at the expense of the number of degrees of freedom in the dataset, and may therefore result in an estimate (weighted mean) with larger errors (see Equation (3.6)) than its least-square counterpart. As confidence in the mean increases, the weighting function favours a narrower cut-off threshold from one iteration to the next.

4.2 Estimation of transfer functions 69

of freedom. From Equation (3.6), we see that a reduction in the number of degrees of freedom will have a tendency to increase the magnitudes of the error estimates for the transfer functions.

However, the omission of noisy data should lead to an increase in coherence (Equation (3.8)), which tends to reduce the magnitudes of the error estimates (e.g., Equation (4.19)). A trade-off, therefore, occurs between the reduced number of degrees of freedom versus increasedcoherence (Section 3.5; Figure 3.8), and whether robust processing results in transfer functions with smaller or larger errors will depend on the amount of available data and on the degree of noise contamination.

As already mentioned (Section 4.1), the choice of length of the data window involves a trade-off between the resolution and the error of the calculated transfer function. Data outliers are more easily identified if their power is not averaged over a large number of Fourier coefficients, favouring the use of short time series windows (having, for example, 256 data points) during processing. However, if we have a sample rate of 2 s, then the longest period that can be included in such a window is of the order 500 s. Larger periods are attained through a process ofdecimation, which involves sub- sampling and low-pass filtering the data (to remove periods shorter than the Nyquist period of the decimated data). For example, by sub-sampling every fourth data point, our effective sampling rate becomes 8 s, the Nyquist period is 16 s, and a data window with 256 points now includes periods of the order 2000 s. Decimation can be performed repeatedly until the desired period range is achieved. In some decimation schemes, the digital low-pass filters do not have a steep cut-off. In our example, we might expect our decimated and undecimated datasets to produce transfer functions that overlap at 32 s. In practice, there is often a mismatch between the longest- period transfer functions calculated from the undecimated data and the shortest-period transfer functions calculated from the deci- mated data: the period at which the two transfer functions overlap is often shifted towards the middle of the period range spanned by the decimated data owing to inadequate low-pass filtering.

Robust processing can help to discriminate against rare, but powerful, source-field heterogeneities of short spatial scales that sometimes occur even at mid-latitudes. Robust processing methods are now widely adopted as an integral part of MT data reduction, having been shown to yield more reliable estimates of MT transfer functions than least-square processing methods in many circum- stances. The flowchart shown in Figure 4.4 summarises the steps involved in processing MT data.

70 Data processing

4.3 ‘Upwards’ and ‘downwards’ biased estimates

In Section 4.2 we explained how components of the impedance tensor can be isolated from inexact measurements of the electromagnetic fields using cross-power spectral densities. From Equations (4.16), we can find six pairs of simultaneous equations containing any given component of the impedance tensor, and each of these simultaneous equations can be solved to give six mathematical expressions for each impedance tensor component (Simset al., 1971). For example, (in the absence of remote reference data (Section 4.2)) the imped- ance tensor componentZ_ijcan be written as:

Z_ij¼ hH~iE~_iihE~iH~_ji hH~iH~_jihE~iE~_ii

hH~_iE~_iihH~_jH~_ji hH~_iH~_jihH~_jE~_ii (4:22a) or

Z_ij¼ hH~_iE~_jihE~_iH~_ii hH~_iH~_iihE~_iE~_ji

hH~_iE~_jihH~_jH~_ii hH~_iH~_iihH~_jE~_ji (4:22b)

Compute

MT TRANSFER FUNCTIONS RAW DATA Local site

Calibration

Frequency averaging

Remote reference

Yes

Stack ‘Clean’ spectra No No Yes

Remote reference site (optional)

Decimation to obtain longer periods Visual inspection

of time series Selection of time windows Fourier transform time windows Compute complex Fourier coefficients

Choose evaluation frequencies

Cleaning (coherence sorting/

robust weighting, etc.) Better trade-off between omitting ‘noisy’

data and preserving number of degrees of freedom required/possible?

Figure 4.4Flowchart

summarising the steps involved in processing MT data.

4.3 ‘Upwards’ and ‘downwards’ biased estimates 71

or

Zij¼ hH~_iE~_iihE~_iH~_ii hH~_iH~_iihE~_iE~_ii

hH~_iE~_iihH~_jH~_ii hH~_iH~_iihH~_jE~_ii (4:22c) or

Z_ij¼ hH~_iE~_iihE~_iE~_ji hH~_iE~_jihE~_iE~_ii

hH~_iE~_iihH~_jE~_ji hH~_iE~_jihH~_jE~_ii (4:22d) or

Z_ij¼ hH~_iH~_iihE~_iH~_ji hH~_iH~_jihE~_iH~_ii

hH~_iH~_iihH~_jH~_ji hH~_iH~_jihH~_jH~_ii (4:22e) or

Zij¼ hH~_iE~_jihE~_iH~_ji hH~_iH~_jihE~_iE~_ji

hH~_iE~_jihH~_jH~_ji hH~_iH~_jihH~_jE~_ji: (4:22f )

For the 1-D case, there should be no correlation between the perpendicular electric fields,EiandEj, nor between the perpendicu- lar magnetic fieldsH_iandH_j, whilst parallel electric and magnetic fields should also be uncorrelated, i.e.,hE~iE~_ji,hE~iH~_ii,hE~jH~_ji, and hH~_iH~_jitend to zero. As such, stable solutions cannot be determined readily from Equations (4.22a) and (4.22b).

Measured electric and magnetic fields are always likely to contain noise. Therefore, we can expand each component of the observed electromagnetic fields as the sum of the natural field plus noise. For example,

E_xOBS ¼E_xþE_xNOISE H_xOBS¼H_xþH_xNOISE

: (4:23)

Assuming that the noise is incoherent (i.e.,ExNOISE,HxNOISE, etc., are random and independent of the natural electromagnetic fields and of each other) then the following auto-power density spectra can be generated from Equations (4.23):

hE_xOBSE_x

OBSi ¼ hE_xþE_xNOISEihE_xþE_x

NOISEi

¼ hExE_xi þ2hExExNOISEi

|fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}

¼0

þhExNOISEE_xNOISEi

¼ hE_xE_xi þ hE_xNOISEE_x

NOISEi:

(4:24)

72 Data processing

Similarly,

hE_yOBSE_y

OBSi ¼ hE_yE_yi þ hE_yNOISEE_y

NOISEi (4:25)

hHxOBSH_x

OBSi ¼ hHxH_xi þ hHxNOISEH_x

NOISEi (4:26)

hH_yOBSH_y

OBSi ¼ hH_yH_yi þ hH_yNOISEH_y

NOISEi: (4:27)

Meanwhile, the cross-spectral density estimates are free of noise if averaged over many samples e.g.,

hExOBSH_y

OBSi ¼ hExþE_xNOISEihH_yþH_y

NOISEi

¼ hExH_yi þ hExH_y

NOISEi

|fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}

¼0

þ hExNOISEH_yi

|fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}

¼0

þ hExNOISEH_y

NOISEi

|fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}

¼0

¼ hE_xH_yi: ð4:28Þ

For the simplest case of a 1-D Earth, Equations (4.22c) and (4.22d), which give stable estimates forZ_ij provided that the incident fields are not highly polarised, may, therefore, be expanded as:

Z_ijOBS¼Z_ij 1þhEiNOISEE_iNOISEi hEiE_ii

: (4:29)

Equat ions (4.2 2e) and (4.22f ) also give stabl e estimat es for Zij

provided that the incident fields are not highly polarised, and may be expanded as:

Z_ijOBS¼Z_ij 1þhHjNOISEH_j

NOISEi hHjH_ji

!1

: (4:30)

From Equations (4.29) and (4.30), we see that estimates ofZij

based on Equations (4.22c) and (4.22d) are biased upwards by incoherent noise in the measured electric fields, whilst estimates of Zijbased on Equations (4.22e) and (4.22f) are biased downwards by incoherent noise in the measured magnetic fields. The degree of bias depends on the signal-to-noise ratio – the more power present in the noise spectra, the greater the bias toZij.

In summary, of the six mathematical expressions forZij, two are biased downward by any random noise present in the magnetic field components, but are unaffected by random noise superimposed on the electric signal, two are biased upwards by random noise imposed

4.3 ‘Upwards’ and ‘downwards’ biased estimates 73

on the electric field components, but are unaffected by any random noise present in the magnetic field components, and the remaining two are unstable and should be avoided.

One method of reducing the effects of this type of biasing of impedance tensor estimates is to incorporate remote reference data (Section 4.2). Another possibility is to calculate a weighted average (Z^AV_ij ) of the up (Z^U_ij) and down (Z^D_ij) biased estimates (e.g., Jones 1977) normalised by the squares of their associated errors (Z^U_ij and Z^D_ij):

Z^AV_ij ¼ Z^U_ij

Z^U_ij2þ Z^D_ij Z^D_ij2

0 B@

1

CA 1

Z^U_ij2þ 1 Z^D_ij2

0 B@

1 CA

1

: (4:31)