A parade of general models and their misfit measures

In the 1960s, 70s and 80s, the initial 1-D (layered-Earth) MT model was superseded by a series of general conductivity models which allowed for more structure than a 1-D model, but not for complete three-dimensionality. The first of these general models was suggested by Cantwell (1960), who replaced scalar impedance with a rank 2 impedance tensor Z that links the two-component horizontal electric and magnetic variational fieldsEandH:

E_x E_y

¼ Z_xx Z_xy Z_yx Z_yy

H_x H_y

: (5:8)

In the layered-Earth model considered by Tikhonov (1950, reprinted 1986), Cagniard (1953) and Wait (1954) the impedance tensor would be:

E_x E_y

¼ 0 Z_n Z_n 0

H_x H_y

(5:9)

in any co-ordinate system (because parallel components ofEandH should be uncorrelated). A rotationally invariant dimensionless misfit measure that can be used to check whether the layered-Earth model described by Equation (5.9) is appropriate for measured data is:

X¼D²₁þS²₂

=D²₂: (5:10)

Ifis significantly larger than the relative errorD2=jD2j, then the data require a more complex model.

In the case of a 2-D conductivity structure, (e.g., two quarter- spaces with different layered structures (Figure 2.5), or a dyke) the impedance tensor can – in an appropriate co-ordinate system – be reduced to

E_x E_y

¼ 0 Z_xy Z_yx 0

H_x H_y

(5:11)

withZ_xyandZ_yxbeing the different impedances of two decoupled systems of equations describing induction with electric fields

5.3 A parade of general models and their misfits 83

parallel and perpendicular, respectively, to theelectromagnetic strike of the 2-D structure (as, for example, described in Section 2.6).

Measured data rarely have zero diagonal impedance tensor elements in any co-ordinate system. This may be a result of:

(i) data errors imposed on a truly 1-D or 2-D inductive response;

(ii) coupling of the regional 1-D or 2-D inductive response with localised, small-scale (relative to the scale of observation), 3-D heterogeneities that cause non-inductive distortion effects; or

(iii) 3-D induction effects.

If departures from the ideal situation (in which a rotation angle can be found that completely annuls the diagonal elem- ents of the impedance tensor) arise owing to extraneous noise, then it may be appropriate to compute an electromagnetic strike as the angle,, in Equation 5.4 that maximises the off-diagonal elements and minimises the diagonal elements of the impedance tensor (Swift (1967, reprinted 1986)). In practice, there are numerous conditions that may be applied in order to determine , and thus retrieve the two principal, off-diagonal impedances for this simple 2-D case: e.g., maximising jZxy2Dj or jZyx2Dj, minimising jZxx2Dj or jZyy2Dj, maximising jZxy2DþZyx2Dj, mini- misingjZxx2DþZyy2Dj, maximisingjZxy2D

2j þ jZyx2Dj², or minimis- ing jZxx2Dj²þ jZyy2Dj². For a truly 2-D structure, all of these criteria should yield the same principal directions (although, in practice, they often do not!)

For example, the angle, , at which jZxxðÞj²þ jZyyðÞj² is minimised is obtained by satisfying the condition that

@

@jZ_xxðÞj²þZ_yyðÞ²

¼0: (5:12)

The condition expressed in Equation (5.12) is satisfied for

tan 4¼ 2ReðS⁰₂D⁰₁Þ D⁰₁

²S⁰₂²: (5:13)

The second derivative of jZxxðÞj²þZyyðÞ² also needs to be considered, in order to confirm that the diagonal components are minimisedby a rotation through angle. Otherwise, a maximum is found and

!þ45 (5:14)

(Swift, 1967, reprinted 1986; Vozoff, 1972).

84 Dimensionality and distortion

The computed strike direction contains a 908 ambiguity, because rotation by 908 only exchanges the location of the two principal impedance tensor elements within the tensor:

90

0 Z_xy Z_yx 0

^T

90¼ 0 1

1 0

0 Z_xy Z_xy 0

0 1

1 0

¼ 0 Z_yx Zxy 0

: (5:15)

Therefore, electromagnetic strikes of and 908 cannot be separated using a purely mathematical model.

The early availability of numerical solutions for induction in 2-D structures (e.g., Jones and Pascoe, 1971) contributed to the widespread use of the Swift model. Swift (1967, reprinted 1986) also provided a rotationally invariant misfit parameter referred to asSwift skew,,:

¼jS₁j=jD₂j; (5:16) which provides an ‘adhoc’ indication as to the appropriateness of applying the Swift model to measured data (see also Vozoff, 1972).

The errors of theSwift strikeand Swift skew are

ðtanÞ ¼ 2S₂D₁ D₁ j j²jS₂j²

!2

þ 2D₁S₂ D₁ j j²jS₂j²

!2

2 4 8<

:

þ 2<eðS₂D₁Þ D₁ j j²jS₂j²

2

0 B@

1 CA

23 75

D₁D₁

ð Þ²þðS₂S₂Þ²

9

=

;

1 2=

(5:17)

¼ S₁ D₂

2

þ D₂S₁ D²₂

2

" #1=2

; (5:18)

where the error functions denoted byare real.

If distortions of the type discussed in (ii) above are present, then the electromagnetic strike determined from the Swift method is unlikely to lead to satisfactory unmixing of the principal imped- ances, and a more complex model will be required.

In Section 2.9, we discussed how the dimensionality of a con- ductivity anomaly depends on the scale at which we observe it (Figure 2.11). When the electromagneticskin depthbecomes several times greater than the dimensions of the anomaly, its electromagnetic

5.3 A parade of general models and their misfits 85

response becomes inductively weak, but it continues to have a non- inductive response (commonly termed ‘galvanic’). In an early con- tribution of 3-D modelling to our understanding of non-inductive distortion, Wannamaker et al. (1984a) demonstrated this type of behaviour arising from a 3-D body embedded in a layered Earth.

Electromagnetic data containing galvanic effects can often be described by a superimposition or decomposition model in which the data are decomposed into a non-inductive response owing to multi-dimensional heterogeneities with dimensions significantly less than theinductive scale lengthof the data (often described aslocal), and a response owing to an underlying 1-D or 2-D structure (often described as regional; Figure 5.3). In such cases, determining the electromagnetic strike involvesdecomposingthe measured impedance tensor into matrices representing the inductive and non-inductive parts. The inductive part is contained in a tensor composed of compo- nents that have both magnitude and phase (i.e., its components are complex), whereas the non-inductive part exhibits DC behaviour only, and is described by a distortion tensor, the components of which must berealand thereforefrequency independent.

Larsen (1975) considered the superposition of a regional 1-D layered-Earth model and a small-scale structure of anomalous con- ductance at the Earth’s surface (Figure 5.4). If the size of the anomaly is small compared to the penetration depth, p, of the

Skin depth >>

dimensions of

2ρ

ρ1 ρ2

1-D INDUCTION 2-D INDUCTION 3-D INDUCTION

GALVANIC DISTORTION Period

Skin depth

}

^Increasing

MT I site II

IV III Figure 5.3A near-surface 3-D

conductor with resistivity2

appears 1-D for MT sounding periods that are sufficiently short to be contained within the conductor, but as the sounding period lengthens, edges I and II are detected and the MT response is one of 2-D induction. Then, as the inductive scale length increases sufficiently to image edges I–IV, the finite strike of the

conductor is revealed by a 3-D inductive response. Owing to charges at the boundaries, galvanic effects are also produced, and at periods for which the electromagnetic skin depth is significantly greater than the dimensions of the near-surface inhomogeneity, the galvanic response dominates over the inductive response.

86 Dimensionality and distortion

electromagnetic field, then the impedance tensor associated with Larsen’s general model can be decomposed as:

E_x E_y

¼C 0 Z

Z 0

H_x H_y

¼ c₁₁ c₁₂ c₂₁ c₂₂

0 Z

Z 0

H_x H_y

; (5:19)

whereZis the regional impedance of Cagniard’s (1953) 1-D model and C is a real distortion tensor, which describes the galvanic (rather than inductive) action of the local scatterer on the electric field. Equation (5.19) contains sixdegrees of freedom– four real distortion parameters and a complex impedance – whereas meas- ured data that can be explained by this model provide only five degrees of freedom – four amplitudes and one common phase.

Therefore, the absolute amplitude of the impedance cannot be separated from the distortion parameters. This is equivalent to stating that static shift factors cannot be mathematically deter- mined from a measured impedance. However, moderate distortion

x

y

z

σ1

σ2

σ3

τn

τa

(a)

τn

σ1

σ2 σ3 σ4

τa

x

y

z (b)

Figure 5.4(a) Larsen’s (1975) superimposition model (Equation (5.19)), in which a small-scale, near-surface conductance anomaly,_a, scatters electric fields, distorting the measured impedance away from that of the regional 1-D layered model in which_ais embedded. (b) Bahr’s (1988) superimposition model (Equation (5.22)) in which a small-scale, near-surface conductance anomaly,a, scatters electric fields, distorting the measured impedance away from that of the regional 2-D model in which_ais embedded.

(Subsequent testing of the superimposition hypothesis on measured data has

demonstrated that (where the hypothesis is applicable) the distorting body is not necessarily surficial).

5.3 A parade of general models and their misfits 87

of synthetic and measured impedance amplitudes can be analysed with simple 3-D models (Park, 1985).

Can measured data be explained with Larsen’s (1975) model? If so, then all elements of the measured impedance should have the same phase. The difference between the phases of two complex numbers xandycan be determined using the commutator:

x; y

½ ¼Imðy xÞ

¼RexImyReyImx: (5:20)

Hence, a rotationally invariant dimensionless misfit measure for the Larsen (1975) model has been proposed as:

¼ðj½D1; S2j þj½S1;D2jÞ¹⁼²=jD2j: (5:21) For some time it was supposed that the two complementary models of Swift (Equation (5.11)) and Larsen (Equation (5.19)) would be adequate to describe measured impedances. That is to say, it was expected that a co-ordinate system should exist for which either (i) the off-diagonal components (ZxyandZyx) of the impe- dance tensor would have different phases (due to different conduct- ivity structures along and across strike), and the diagonal components (ZxxandZyy) would be negligible; or (ii) all elements of the impedance tensor would exhibit the same phase, but the diagonal components would be non-zero. However, as demonstrated by Ranganayaki (1984), MT phases can depend strongly on the direction in which the electric field is measured, and the existence of a large class of measured impedances having both non-vanishing diagonal components and two different phases, such that they con- curred with neither the Swift model nor the Larsen model, led Bahr (1988) to propose a more complete superimposition (decomposi- tion) model. In this model, multi-dimensional heterogeneities with dimensions significantly less than the inductive scale length of the data are superposed on a regional 2-D structure, and the data are decomposed into a ‘local’, non-inductive response (galvanic), and a ‘regional’, inductive response. For data aligned in the (x⁰, y⁰) co-ordinate system of the regional 2-D structure, the impedance tensor can then be expanded as:

E_x Ey

!

¼C

0 Z_n;x0y⁰

Zn;y⁰x⁰ 0

! H_x Hy

!

¼ c₁₂Z_n;y0x⁰ c₁₁Z_n;x0y⁰

c22Z_n;y0x⁰ c₂₁Z_n;x0y⁰

! H_x H_y

! :

(5:22)

88 Dimensionality and distortion

Within each column, only one phase occurs, because the assump- tion of galvanic distortion requires that the elements of the distortion tensor,C, must be real and frequency independent. In an arbitrary co-ordinate system, however, the phases of the two regional imped- ancesZ_n;xyandZ_n;yxwill be mixed, because in this case the impedance tensor elements are linear combinations of Zn;xy and Zn;yx. In an arbitrary co-ordinate system, we have:

Z¼

C Z

2D^T

: (5:23)

The condition that, in the co-ordinate system of the regional strike, the tensor elements in the columns of the tensor should have the same phase, i.e,

ReðZ_xxÞ

ImðZ_xxÞ¼ReðZyxÞ

ImðZ_yxÞ)ReðZ_xxÞ

ReðZ_yxÞ¼ImðZ_xxÞ

ImðZ_yxÞ (5:24) leads to an equation for the rotation angle,:

Asinð2Þ þBcosð2Þ þC¼0;

where

A¼ ½S₁;D₁ þ ½S₂;D₂ B¼ ½S₁;S₂ ½D₁;D₂ C¼ ½D1;S₂ ½S1;D₂:

(5:25)

The two solutions:

tan1;2¼ nðBþCÞ=ðBCÞ þ½A=ðBCÞ²o1=2

A=ðBCÞ (5:26) lead to a co-ordinate frame where the same phase condition is fulfilled either in the left or in the right column of the impedance tensor. Only if the commutator C is negligible, does a unified co-ordinate frame exist for which the same phase condition is fulfilled in both columns of the impedance tensor. In this case, the rotation angle,, is found from:

tan¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þ A

B s 2

A

B or tanð2Þ ¼B=A: (5:27)

The electromagnetic strike that is recovered using the Bahr model is often referred to as thephase-sensitive strike.

For cases in which no rotation angle can be found for which the phases in the respective columns of the impedance tensor are equal, Bahr (1991) proposes minimising the phase differencebetween the elements of a given column such that:

Z¼ c₁₂Z_yxreⁱ c₁₁Z_xyr c22Z_yxr c₂₁Z_xyreⁱ

^T: (5:28)

5.3 A parade of general models and their misfits 89

In this case,

ReðZ_x0x⁰ÞcosþImðZ_x0x⁰Þsin

ReðZy⁰x⁰Þ ¼ReðZ_x0x⁰ÞsinþImðZ_x0x⁰Þcos ReðZy⁰x⁰Þ :

(5:29)

Equation (5.29) can be solved forand(Bahr, 1991; Pra´cser and Szarka, 1999⁶) to yield:

tan 2 _1;2

¼1 2

B₁A₂þA₁B₂þC₁E₂

ð Þ

A1A2C1C2þC1F2

ð Þ

1 4

B₁A₂þA₁B₂þC₁E₂

ð Þ²

A₁A₂C₁C₂þC₁F₂

ð Þ² ðB₁B₂C₁C₂Þ A₁A₂C₁C₂þC₁F₂

ð Þ

" #1=2

;

(5:30)

where

A¼A₁þA₂¼ ½S ₁;D₁ þ ½S2;D₂

cosþðfS₁;D₁g þfS₂;D₂gÞsin B¼B1þB2¼ ½Sð 1;S2 ½D1;D2ÞcosþðfS1;S2g fD1;D2gÞsin C¼C₁þC₂¼ ½Dð ₁;S₂ ½S₁;D₂ÞcosþðfD₁;S₂g fS₁;D₂gÞsin E¼E₂¼ðfS₂;S₂g fD₁;D₁gÞsin; F₂¼2fD₁;S₂gsin;

and

¼tan¹ C₁

A2sin 2ð Þ þB2cos 2ð Þ

ð Þ

; (5:31)

where the commutator ½x;y is defined by Equation (5.20) and x;y

f g ¼ReðyxÞ. Bahr (1991) refers to this extension of the super- imposition model as thedelta () technique.

Bahr (1988) proposed a rotationally invariant parameter termed phase-sensitive skewas an ad hoc measure of the extent to which an impedance tensor can be described by Equations (5.22) or (5.28):

¼ ffiffiffiffi pC

=jD₂j ¼ðj½D₁;S₂ ½S₁;D₂jÞ¹⁼²=jD₂j: (5:32) For <0.1, Bahr suggested that Equation (5.22) is the appro- priate model, whereas for 0.1<<0.3 Equation (5.28) might be more appropriate. However, see our warnings concerning skew in Section 5.7.

Another technique that is routinely used to solve for the electro- magnetic strike using the decomposition hypothesis is an inverse technique proposed by Groom and Bailey (1989). Central to the decomposition hypothesis, whether solved using the Bahr formulation

6Equation (5.30) is the correct solution of Equation (5.29) provided by Pra´cser and Szarka (1999). It should replace the incorrect solution by Bahr (1991).

90 Dimensionality and distortion

or the Groom and Bailey formulation is the requirement that the distortion tensor should be real and frequency independent. As for the Bahr technique, electric, rather than magnetic distortion of the impedance tensor is considered to be of greatest significance in the Groom and Bailey decomposition model.

In Groom and Bailey’s decomposition technique, separation of the ‘localised’ effects of 3-D current channelling from the ‘regional’ 2-D inductive behaviour is achieved by factorising the impedance tensor problem in terms of a rotation matrix,

, and a distortion tensor,C, which is itself the product of three tensor suboperators (twist,T, shear,S, local anisotropy,A,⁷and a scalar,g,:

Z¼

C Z

2D^T

where C¼gT S A: (5:33)

Groom and Bailey illustrate the need for the four independent parameters in the distortion tensor factorisation by considering a scenario in which MT data are collected at the centre of an elliptical swamp, which presents a highly conductive surface region, encom- passed by an insulating substratum in a moderately conductive region (Figure 5.5). The presence of the swamp causes the telluric currents to be twisted to its local strike. This clockwise rotation of the telluric vectors is contained in the twist tensor,T. The swamp also causes splitting of the principal impedances by generating dif- ferent distortion-related stretching factors. This anisotropic effect is contained in the anisotropy tensor,A. Meanwhile, the shear tensor, S, stretches and deflects the principal axes so that they no longer lie orthogonal to each other. WhereasAdoes not change the directions of telluric vectors lying along either of the principal axes,Sgener- ates the maximum angular deflection along these principal axes. In the case of strong current channelling, the direction of the vari- ational electric field does not depend on the direction of the mag- netic field at all. Distortions manifest as non-orthogonal telluric vectors may also arise owing to inaccurate alignment of telluric dipoles during site setup.

Equation (5.33) presents an ill-posed problem as it contains nine distinct unknown parameters – the regional azimuth, the four components of the distortion tensor and the two complex regional impedances – whereas the complex elements of the measured imped- ance tensor provide only eight known parameters. Therefore, the

7This is a distortion effect and does not have the same physical meaning as

‘electrical anisotropy’ as proposed, for example, by Kelletet al. (1992). Electrical anisotropy is discussed with reference to case studies in Chapter 9.

5.3 A parade of general models and their misfits 91

decomposition model as posed in Equation (5.33) does not have a unique solution. However, a unique solution does exist if the linear scaling factor,g, and anisotropy,A, are absorbed into an equivalent ideal 2-D impedance tensor that only differs fromZ2-Din that it is scaled by real, frequency-independent factors:

Z⁰

2-D¼gA Z

2-D: (5:34)

This transformation leaves the shapes of the apparent resistivity and impedance phase curves unchanged, but the apparent resistivity curves will be shifted by unknown scaling factors. Once again (cf.

Larsen’s model), this is equivalent to stating that there is no way of determining static shift factors mathematically from a measured impedance.

Although a unique solution of the decomposition model exists theoretically, in practice measured data, which contain noise and departures from the model, will never yield a perfect fit to the decomposition model in any co-ordinate system. Therefore a least- square (Section 4.2) fitting procedure is employed. In Groom–Bailey decomposition, a misfit parameter between the measured data,Z

ij, θt

(a)

x^I

y^I x

y

(b) ^x

y

x x^I

y y^I

x

y^I x^I

y Twist:T= ¹

√^1+t2

1 –t t 1

Shear:S= ¹

√^1+e2

1 e e 1

Anisotropy:A= ¹

√^1+s2

1+s 0 0 1–s

Figure 5.5(a) A contrived scenario in which MT data are collected at the centre of a conductive swamp (black) that is encompassed by a moderately conductive region (grey), and an insulator (white). tdenotes the local strike of the swamp, which

‘twists’ the telluric currents.

The anomalous environment also imposes shear and anisotropy effects on the data.

(b) Distortion of a set of unit vectors by twistT, shear,S, and ‘anisotropy’,A, operators, which are parameterised in terms of the real valuest,eand s, respectively. (Redrawn from Groom and Bailey, 1989.)

92 Dimensionality and distortion

data errors,_ij, and data modelled according to the 2-D hypothesis, Zij;mis suggested as:

²¼1 4

P²

j¼1

P²

i¼1

Z^_ij;mZ_ij

²

P²

i¼1

P²

j¼1

_ij ²

: (5:35)

3-D induction effects are not considered directly in the decom- position parameterisation, but their presence can be detected by investigating distortion parameters and misfit parameters. In the presence of 3-D induction, the computed distortion parameters will exhibit frequency dependence, and misfit parameters can be expected to be large. However, note that²is normalised by the errors in the data. So, a changing error structure in different period ranges may influence the misfit.

Decomposition models (Groom and Bailey, 1989; Bahr, 1991;

Chave and Smith, 1994; and Pra´cser and Szarka, 1999) have been applied to many datasets. Figure 5.6 summarises the seven classes of impedance tensor that have been proposed. The parameters (such as regional strike) for the more-complicated models can be unstable if a measured impedance can already be described by a simpler general model. For example, Equation (5.22) will not yield stable results if an impedance tensor can be described by Equation (5.19) (Berdichevsky, 1999). In cases of strong current channelling or 3-D induction, the electromagnetic strike obtained using mathematical decomposition is also unstable.

The decomposition model described in Equation (5.22) was adopted by Kelletet al. (1992), who suggested that the existence of an electromagnetic strike and the impedance phase difference associated with it can be a consequence of electrical anisotropy in a particular depth range. Kelletet al.(1992) proposed a physical model that comprises a superposition of a surface scatterer and a 1-D electrically anisotropic regional structure (Figure 5.7). Although developed for a limited target area, Kelletet al.’s model may explain why decomposition models can apparently be used to describe so many different datasets – given that a 1-D anisotropic structure is mathematically equivalent to a 2-D structure. An example of the impedance phase split created by an anisotropic structure is pre- sented in Figure 5.8. The concept of lower crustal electrical aniso- tropy in connection with the impedance tensor decomposition model was used by Joneset al. (1993) and Eisel and Bahr (1993) to explain the data shown in Figure 5.8, with resistivity ratios of up to 60 between maximum and minimum directions of resistivity in

5.3 A parade of general models and their misfits 93

the horizontal plane. Bahret al. (2000) suggested a general ‘aniso- tropy test’ based on the impedance phases present in electromag- netic array data, and showed that this test could distinguish between the model of crustal anisotropy and the model of an isolated con- ductivity anomaly.

In the course of applying decomposition techniques, care should be taken to consider the appropriateness of the technique Class 1a : Simple 1-D model (Cagniard, 1953), Equation (5.9)

Class 1b : Simple 2-D model (Swift, 1967), Equation (5.11)

Class 2 : Regional 1-D model with galvanic distortion (Larsen, 1977), Equation (5.19) Class 3 : 2-D model with static shift (Bahr, 1991), Equation (5.22)

Class 4 : Regional 2-D model in ‘twisted’ co-ordinates (Bahr, 1991), Equation (5.22) Class 5a : 2-D superimposition model (Bahr, 1988), Equation (5.22)

Class 5b : 2-D Extended superimposition ( -) model (Bahr, 1991), Equation (5.28) Class 6 : Regional 2-D model with strong local current channelling (Bahr, 1991) Class 7 : Regional 3-D induction model

d Yes

STOP START

Class 4

Class 6

Class 5b

Class 1a Class

1b

Class 7

Class 2

Class 5a

Class 3 S<0.05

m<0.05 k<0.1

h<0.3 h<0.1 No

No

Yes No

No

No No

Yes No

Yes Yes

Yes Yes

Yes Yes

b1<5 and^o <20^o

< 5 and < 20 b

b b

2

2 1

or

o o

Class

6 -b1+b2~~90^o -b1+b2~~90^o

STOP STOP

STOP

STOP

STOP

b1=b2

Stable, frequency- independent strike

No Stable, frequency- independent strike STOP

Yes No Yes STOP

STOP No

No

START THINKING!

Figure 5.6Flowchart summarising characterisation and model parameterisation for different classes of the impedance tensor.

94 Dimensionality and distortion