P. Schmoldt, PhD - MTNet - DIAS
P. Schmoldt, PhD - MTNet - DIAS P. Schmoldt, PhD - MTNet - DIAS
6. Using magnetotellurics to gain information about the Earth mean cross-spectral density (S xy), which in turn is the product of the respective FT’s X( f ) and Y( f ), i.e. C 2 xy( f ) = 〈S xy( f )〉 2 〈S xx( f )〉〈S yy( f )〉 = |〈X ∗ ( f )Y( f )〉| 2 〈X ∗ ( f )X( f )〉〈Y ∗ ( f )Y( f )〉 (6.3) where the asterisk denotes the complex conjugate of the function. Coherence weighting is implicitly applied during the impedance estimation using the remote reference method, in which modern processing schemes also consider partial and multiple coherence as indicators for the quality of an impedance estimate. To illustrate the application of this procedure to an MT dataset consider the explicit form of Equation 3.34 for the electric field component in x-direction (Ex), i.e. Ex = ZxxHx + ZxyHy (6.4) where both, the electric and magnetic channels Ex and Hy are assumed here to contain noise. Multiplication of this Equation with the complex conjugate of either the electric or magnetic field component in x-direction (R ∗ x) or in y-direction (R ∗ y) yields and 〈ExR ∗ x〉 = Zxx〈HxR ∗ x〉 + Zxy〈HyR ∗ x〉 (6.5) 〈ExR ∗ y〉 = Zxx〈HxR ∗ y〉 + Zxy〈HyR ∗ y〉, (6.6) respectively. Combining these two equations results in an estimate of the Zxy component Zxy = 〈ExR∗ y〉〈HxR∗ x〉 − 〈ExR∗ x〉〈HxR∗ y〉 〈HxR∗ x〉〈HyR∗ y〉 − 〈HxR∗ y〉〈HyR∗ . (6.7) x〉 Expressions for all components of the impedance tensor can be derived in a similar manner, therefore the general form can be stated as Zi j = 〈EiR∗ ∗ ∗ j 〉〈HkRk 〉 − 〈EiRk 〉〈HkR∗ j 〉 , (6.8) DET with DET = 〈HxR ∗ x〉〈HyR ∗ y〉 − 〈HxR ∗ y〉〈HyR ∗ x〉, and i, j, k ∈ [x, y], with k j. Whether processing with the electric or the magnetic component as reference provides superior results depends on the type of noise and its effect on each channel. The magnetic component is commonly assumed to be less contaminated by noise and usually chosen as remote, however the choice is often a rather subjective one. To use a combination of remote references from different components or stations can often be useful. 112 For the case of a perfectly 2D subsurface and according rotation of the magnetic field
data, Hx and Hy are uncorrelated and Equation 6.8 reduces to Zi j = 〈EiH ∗ j 〉 〈H jH ∗ j 〉, 6.2. Processing of magnetotelluric data (6.9) in situations where the magnetic field is used as remote reference. Separating the electric and magnetic components into a term containing the noise (NEi and NH ) and a noise free j term (Ei f and H j ) yields f Zi j = 〈Ei f H∗ j 〉 + 〈NEi H∗ j 〉 〈H j f H∗ j 〉 + 〈NH jH∗ j 〉. (6.10) In Equation 6.10 it is shown that the effect of noise on the impedance estimate depends on the correlation between the noise in either the electric or magnetic channel and the chosen remote reference channel. For dominant correlation between the noise in the electric channel and the remote reference, the term 〈NEi H∗ j 〉 will cause an overestimation of the impedance, whereas, in the opposite case where 〈NH jH∗ j 〉 dominates, the impedance will be underestimated. Estimates of the magnetic transfer function in the presence of disturbances are derived accordingly, i.e. 〈HzR ∗ 〉 = Tx〈HxR ∗ 〉 + Ty〈HyR ∗ 〉 (6.11) with R∗ denoting the respective remote reference. Therefore transfer function components are estimated as Ti = 〈HzR∗ i 〉〈H jR∗ j 〉 − 〈HzR∗ j 〉〈HiR∗ i 〉 , (6.12) DET with DET as defined for Equation 6.8, and i, j ∈ [x, y] with j i. Robust processing methods A processing method is considered robust when it is relatively insensitive to the presence of a moderate amount of bad data [Jones et al., 1989], and thus able to cull out a superior set of estimates from a contaminated data set. Robust processing methods have been adapted for MT processing and their application was discussed, among others, by Egbert and Booker [1986]; Chave et al. [1987]; Chave and Thomson [1989]; Larsen et al. [1996]; Smirnov [2003]. Common applications of robust processing in MT include bounded influence estimator, M-estimator [Huber, 1981], or Jack-knife processing and iterative rejection of estimates to either increase the coherence of the estimates or decrease the variance (or standard deviation) of the resulting impedance estimate. Errors of the resulting estimates are then calculated on a statistical basis using Bootstrap analysis. The crucial point of a processing algorithm, i.e. its robustness, is the breakdown point ε ∗ , which describes the maximal fraction of erroneous data that can be handled by the algorithm [Hampel et al., 1986]. Certainly the optimal breakdown point is 50 %, common robust algorithms, using the M-estimator and linear regression, approach 30 %; LS 113
- Page 97 and 98: 4.1. Types of distortion the use of
- Page 99 and 100: 4.2. Dimensionality Fig. 4.12.: The
- Page 101 and 102: 1D 2D local 3D/1D 3D/2D regional 4.
- Page 103 and 104: 4.3. General mathematical represent
- Page 105 and 106: 4.4. Removal of distortion effects
- Page 107 and 108: Parameter Geoelectrical application
- Page 109 and 110: 4.4. Removal of distortion effects
- Page 111 and 112: 4.4.5. Caldwell-Bibby-Brown phase t
- Page 113 and 114: 4.4. Removal of distortion effects
- Page 115: Method Applicability Swift angle 2D
- Page 118 and 119: 5. Earth’s properties observable
- Page 120 and 121: 5. Earth’s properties observable
- Page 122 and 123: 5. Earth’s properties observable
- Page 124 and 125: 5. Earth’s properties observable
- Page 126 and 127: 5. Earth’s properties observable
- Page 128 and 129: 5. Earth’s properties observable
- Page 130 and 131: 5. Earth’s properties observable
- Page 132 and 133: 5. Earth’s properties observable
- Page 134 and 135: 5. Earth’s properties observable
- Page 136 and 137: 5. Earth’s properties observable
- Page 138 and 139: 5. Earth’s properties observable
- Page 140 and 141: 5. Earth’s properties observable
- Page 142 and 143: 6. Using magnetotellurics to gain i
- Page 144 and 145: 6. Using magnetotellurics to gain i
- Page 146 and 147: 6. Using magnetotellurics to gain i
- Page 150 and 151: 6. Using magnetotellurics to gain i
- Page 152 and 153: 6. Using magnetotellurics to gain i
- Page 154 and 155: 6. Using magnetotellurics to gain i
- Page 156 and 157: 6. Using magnetotellurics to gain i
- Page 158 and 159: 6. Using magnetotellurics to gain i
- Page 160 and 161: 6. Using magnetotellurics to gain i
- Page 162 and 163: 6. Using magnetotellurics to gain i
- Page 164 and 165: 6. Using magnetotellurics to gain i
- Page 168 and 169: Part II Geology of the study area I
- Page 170 and 171: 7. Geology of the Iberian Peninsula
- Page 172 and 173: 7. Geology of the Iberian Peninsula
- Page 174 and 175: 7. Geology of the Iberian Peninsula
- Page 176 and 177: 7. Geology of the Iberian Peninsula
- Page 178 and 179: 7. Geology of the Iberian Peninsula
- Page 180 and 181: 7. Geology of the Iberian Peninsula
- Page 182 and 183: 7. Geology of the Iberian Peninsula
- Page 184 and 185: 7. Geology of the Iberian Peninsula
- Page 186 and 187: 7. Geology of the Iberian Peninsula
- Page 188 and 189: 7. Geology of the Iberian Peninsula
- Page 190 and 191: 7. Geology of the Iberian Peninsula
- Page 192 and 193: 7. Geology of the Iberian Peninsula
- Page 194 and 195: 7. Geology of the Iberian Peninsula
- Page 196 and 197: 7. Geology of the Iberian Peninsula
data, Hx and Hy are uncorrelated and Equation 6.8 reduces to<br />
Zi j = 〈EiH ∗ j 〉<br />
〈H jH ∗ j 〉,<br />
6.2. Processing of magnetotelluric data<br />
(6.9)<br />
in situations where the magnetic field is used as remote reference. Separating the electric<br />
and magnetic components into a term containing the noise (NEi and NH ) and a noise free<br />
j<br />
term (Ei f and H j ) yields<br />
f<br />
Zi j = 〈Ei f H∗ j 〉 + 〈NEi H∗ j 〉<br />
〈H j f H∗ j 〉 + 〈NH jH∗ j 〉.<br />
(6.10)<br />
In Equation 6.10 it is shown that the effect of noise on the impedance estimate depends on<br />
the correlation between the noise in either the electric or magnetic channel and the chosen<br />
remote reference channel. For dominant correlation between the noise in the electric<br />
channel and the remote reference, the term 〈NEi H∗ j 〉 will cause an overestimation of the<br />
impedance, whereas, in the opposite case where 〈NH jH∗ j 〉 dominates, the impedance will<br />
be underestimated.<br />
Estimates of the magnetic transfer function in the presence of disturbances are derived<br />
accordingly, i.e.<br />
〈HzR ∗ 〉 = Tx〈HxR ∗ 〉 + Ty〈HyR ∗ 〉 (6.11)<br />
with R∗ denoting the respective remote reference. Therefore transfer function components<br />
are estimated as<br />
Ti = 〈HzR∗ i 〉〈H jR∗ j 〉 − 〈HzR∗ j 〉〈HiR∗ i 〉<br />
, (6.12)<br />
DET<br />
with DET as defined for Equation 6.8, and i, j ∈ [x, y] with j i.<br />
Robust processing methods<br />
A processing method is considered robust when it is relatively insensitive to the presence<br />
of a moderate amount of bad data [Jones et al., 1989], and thus able to cull out<br />
a superior set of estimates from a contaminated data set. Robust processing methods<br />
have been adapted for MT processing and their application was discussed, among others,<br />
by Egbert and Booker [1986]; Chave et al. [1987]; Chave and Thomson [1989]; Larsen<br />
et al. [1996]; Smirnov [2003]. Common applications of robust processing in MT include<br />
bounded influence estimator, M-estimator [Huber, 1981], or Jack-knife processing and iterative<br />
rejection of estimates to either increase the coherence of the estimates or decrease<br />
the variance (or standard deviation) of the resulting impedance estimate. Errors of the<br />
resulting estimates are then calculated on a statistical basis using Bootstrap analysis.<br />
The crucial point of a processing algorithm, i.e. its robustness, is the breakdown point<br />
ε ∗ , which describes the maximal fraction of erroneous data that can be handled by the<br />
algorithm [Hampel et al., 1986]. Certainly the optimal breakdown point is 50 %, common<br />
robust algorithms, using the M-estimator and linear regression, approach 30 %; LS<br />
113
Change language