For anN-layered half-space (Figure 2.3), within every layern we have a diffusion equation (of the form given in Equation (2.14)) containing conductivityn and a solution (cf. Equation (2.17)) of the form:
Exnðqn; !Þ ¼E1nei!qnzþE2nei!þqnz¼anðqn; !Þeqnzþbnðqn; !Þeþqnz; (2:26) withqnbeing defined similarly toq(in Equation (2.19)), but incorp- orating the conductivity,n, of thenth layer. In this case, since each layer has limited thickness,E2n6¼0 becausezcannot be arbitrarily large.
Similarly to Equation (2.26), the magnetic field within thenth layer is given by:
Bynðqn; !Þ ¼qn
i!½anðqn; !Þeqnzbnðqn; !Þeþqnz: (2:27) An hypothetical MT sounding penetrating thenth layer could measureExnandByn. This would allow the following transfer func- tions to be computed:
CnðzÞ ¼ ExnðzÞ
i!BynðzÞ and qn¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi i0n!
p : (2:28)
z0=0
zn – 1
zn
zN – 1
z2
z1
σ1
σ2
σn
σN
Figure 2.3N-layered half-space.
2.5 Induction in a layered half-space 23
By substituting Equations (2.26) and (2.27) into Equation (2.28), we can derive expressions for the transfer functions Cnðzn1Þ and CnðznÞat the top and bottom of thenth layer, respectively.
At the top of thenth layer we have
Cnðzn1Þ ¼ aneqnzn1þbneþqnzn1
qnðaneqnzn1bneþqnzn1Þ: (2:29) and at the bottom of thenth layer
CnðznÞ ¼ aneqnznþbneþqnzn
qnðaneqnznbneþqnznÞ: (2:30) Equation (2.30) can be rearranged to yield a term for the ratio an=bn, which we substitute into Equation 2.29 to give:
Cnðzn1Þ ¼ 1 qn
qnCnðznÞ þtanh½qnðznzn1Þ
1þqnCnðznÞtanh½qnðznzn1Þ: (2:31) The field components are continuous at the transition from the nth to the (n1)th layer, and it follows that the Schmucker–Weidelt transfer function is also continuous so that
CnðznÞ ¼ lim
z!zn0CnðzÞ ¼ lim
z!znþ0Cnþ1ðzÞ ¼Cnþ1ðznÞ: (2:32) Inserting the continuity conditions from Equation (2.32) and substitutingln¼znzn1into Equation (2.31) yields:
Cnðzn1Þ ¼ 1 qn
qnCnþ1ðznÞ þtanhðqnlnÞ
1þqnCnþ1ðznÞtanhðqnlnÞ: (2:33) Equation (2.33) is known as Wait’s recursion formula (Wait, 1954). We can use Wait’s recursion formula to calculate the transfer function at the top of thenth layer if the transfer function at the top of the (n+1)th layer is known. In order to solve Equation (2.33), we must therefore iterate from the transfer function at the top of the lowermost layer (N), which we define to be an homogeneous half- space, such that (from Equation (2.24))
CN¼ 1
qN: (2:34)
Next, we apply Equation (2.33) (N1 times) until we have the transfer function at the surface of the layered half-space, which can be compared to field data.
We now defineapparent resistivityas the average resistivity of an equivalent uniform half-space, (the resistivity of which we calcu- lated from the Schmucker-Weidelt transfer function in Section 2.4 (Equation (2.25)):
24 Basic theoretical concepts
að!Þ ¼ jCð!Þj20!: (2:35) Apparent resistivity is one of the most frequently used param- eters for displaying MT data. We can also calculateafrom syn- thetic data by applying the layered-Earth model represented by Equation (2.33) and Figure 2.3. We leave it as an exercise for the reader to show that becausearepresents an average taken over the whole volume of the half-space that is penetrated, a¼1=1 for penetration depths shallower than z1 (see Figure 2.3), but that a6¼n(wheren¼1=n) when the penetration depth exceeds the thickness of the first layer.
BecauseCis complex, we can also extract animpedance phase.
This is one of the most important MT parameters. The impedance phase,1-D, of our one-dimensional (1-D), layered half-space model can be calculated from
1-D¼tan1ðEx=ByÞ: (2:36) Apparent resistivity and impedance phase are usually plotted as a function of period,T ¼2p=!. The functionsaðTÞandðTÞare not independent of each other, but are linked via the following Kramers–Kroenig relationship (Weidelt, 1972):
ð!Þ ¼p 4!
p ð
1
0
logaðxÞ 0
dx
x2!2: (2:37)
Equation (2.37) states that the functionaðTÞcan be predicted from the functionðTÞexcept for a scaling coefficient,0. The fact that in some two-dimensional (2-D) and three-dimensional (3-D) conductivity distributions the form ofaðTÞis predictable from the impedance phase, whereas the absolute level is not, reflects the
‘distortion’ or ‘static shift’ phenomenon. This is explained in more detail in Chapter 5.
In order to estimate the effect of conductive or resistive layers on the impedance phase, let us consider two idealised two-layer models.
Model I consists of an infinitely resistive layer of thicknessh¼l1, covering an homogeneous half-space with resistivity. Model II consists of a thin layer withconductance(the product of conductiv- ity and thickness) ¼1l1, again overlying an homogeneous half- space with resistivity. For both models, the penetration depth in the top layer can be considered large compared to the thickness of the top layer, but for different reasons: in model I the resistive layer can be thick, because infinite penetration depths are possible, whereas in model II, the top layer is conductive but assumed to be
2.5 Induction in a layered half-space 25
so thin that the electromagnetic field is only moderately attenuated.
Therefore, for both models we have:
q1
j jl11 and; therefore; tanhðq1l1Þ ¼q1l1: Hence, in both models Equation (2.33) reduces to
C1ðz¼0Þ ¼ C2þl1
1þq1C2q1l1: (2:38) In model I, we have 1=21 and therefore jq1C2j 1.
Therefore,
C1¼C2þl1¼C2þh: (2:39) In model II, we have q12¼i!01, l1jC2j and 1l1¼. Therefore,
C1¼ C2
1þi!0C2: (2:40)
(The condition forl1 states thatl1 is so thin that most of the attenuation of the electromagnetic field occurs in the second layer).
In model I – becausehis a real number – only the real part of the complex number C is enlarged. (i.e., the real part of C1 becomes larger than the real part ofC2). A comparison of Equations (2.24) and (2.36) tells us that the phase ofCand the magnetotelluric phase (Equation (2.36)) are linked according to
¼argCþ90: (2:41)
Given that real and imaginary parts of the transfer functionC2
of the homogeneous half-space are equal in magnitude (Equation (2.21)), the magnetotelluric phase (Equation (2.36)) of the homo- geneous half-space is 458. The magnetotelluric phase that is calcu- lated from C1 will therefore be greater than 458 for model I. In model II, real and imaginary parts ofC1are reduced by the attenu- ation of the electromagnetic field in the conducting top layer, and the magnetotelluric phase, , is less than 458. Magnetotelluric phases that are greater than 458are therefore diagnostic of substrata in which resistivity decreases with depth (e.g., Figure 2.4(a)), and lead to a model from whichhcan be determined (Equation (2.39)).
On the other hand, magnetotelluric phases that are less than 458are diagnostic of substrata in which resistivity increases with depth (e.g., Figure 2.4(b)), and lead to a model from which can be determined (Equation (2.40).
A convenient display of data that can be plotted on the same ðzÞ graph as a 1-D model is provided by the –z transform (Schmucker, 1987), wherezis defined by Re(C), andis given by
26 Basic theoretical concepts
¼2acos2 for > 45ðModel IÞ (2:42)
¼a=ð2 sin2Þ for < 45ðModel IIÞ: (2:43) If¼45, Equations (2.42) and (2.43) both yield ¼ a, and the model reverts to the one of an homogeneous half-space (i.e., in the absence of the resistive/conductive layer,h¼0=¼0).
To describe the transfer function associated with a three-layer
‘sandwich’ model consisting of a conductive middle layer with con- ductance and a resistive upper layer of thickness h, Equations (2.39) and (2.40) can be combined to yield
C1¼C2þh¼ C3þh
1þi!0C3; (2:44)
whereC1,C2 andC3are the transfer functions at the top of the uppermost, middle and third layer, respectively. Typical forms of the apparent resistivities and impedance phases as a function of period generated by such a ‘sandwich’ model are depicted in Figure 2.4(c).
Resistivity (Ωm)
Depth (km)
101
101
101
101
101
101 101
101
101
101
101
101
101 101 101 102
102
102
102
102
102
102 102 102 102
102
102 103
103
103
103
103
103
103 103 103 103
103
103 102
102
102
104
104
104 104
104
104 103
103
103
Resistivity (Ωm)
Depth (km)
100
100
100
10–3
10–3
10–3
10–3 10–3 10–3
10–2
10–2
10–2
10–2
10–2 10–2
10–1
10–1
10–1
10–1 10–1 10–1 100
100
100
100
100
100
100 100 100 Apparent resistivity (m)Ω
90
Phase ()o
Period (s) 0
45
Period (s)
Apparent resistivity ()Ωm
90
Phase ()o
Period (s) 0
45
Period (s) Resistivity (Ωm)
Depth (km) Apparent resistivity ()Ωm
90
Phase ()o
Period (s) 0
45
Period (s)
(a)
(b)
(c)
Figure 2.4(a)–(c)
Period-dependent apparent resistivities and impedance phases generated by a layered half-space model in which: (a) resistivity decreases with depth. At the shortest periods the impedance phases are 458, consistent with a uniform half-space model and increase above 458at ~10 Hz, consistent with the decrease in resistivity;
(b) resistivity increases with depth. At the shortest periods the impedance phases are 458, consistent with a uniform half-space model, and decrease below 458at ~10 Hz, consistent with the increase in resistivity;
(c) is incorporated a 10-km- thick high-conductivity layer representing the mid crust. The three models shown in (a), (b) and (c) all terminate with a 5m half-space at 410 km.
2.5 Induction in a layered half-space 27