currents flowing along the modelled profile. A global measure,WG, of the misfit function expressed in Equation (7.8) that is not dominated by sites with very large or very smalli is (Smith and Booker, 1991):
WG¼Xisites
i¼1
Wi
iþmedian: (7:9)
Convergence to the model with the smallest roughness commensur- ate with an acceptable (minimum) misfit requires several iterations (typically 10–20). If a priori knowledge about the presence of sharp discontinuities in resistivity is available, then the penalty for rough- ness can be removed at cells that border on these expected discontinuities (e.g., de Groot-Hedlin and Constable, 1990).
A flowchart of the steps involved in the inversion process is shown in Figure 7.2. It is advisable to experiment with different starting models to ensure that a similar solution is reached independent of the starting model. Smith and Booker (1991) suggest producing a 2-D model by introducing data components over a series of RRI inversions (Figure 7.3). The computation time required for RRI is less than for the Occam inversion scheme, because in the former lateral electric and magnetic field gradients calculated from the previous iteration are incorporated into the forward-modelling stage as an approximation to the new fields, and the forward model responses are computed as successive per- turbations on a string of 1-D models that represent the structure beneath individual sites. Another popular 2-D inversion scheme, which is simple to install and implement has been programmed by Mackieet al.(1997).
Some inversion algorithms (e.g., Smith and Booker, 1991;
de Groot-Hedlin, 1991) allow static shiftparameters to be calcu- lated simultaneously with the modelling procedure by favouring incorporation of surface structure at adjacent sites that minimises horizontal conductivity gradients at greater depths in the model (see Section 5.8).
anisotropic layer between 12 and 22 km depth, embedded within a layered half-space. The synthetic data containing anisotropy (Figure 7.4(b)) were generated using the 2-D forward model code from Wannamakeret al.(1986), and were then inverted using a 2-D inversion routine from Mackieet al.(1997). The data (Figure 7.4(b)) from all twelve hypothetical sites are equivalent, because they all image exactly the same 1-D layered structure with an anisotropic layer composed of a regular sequence of alternating dykes having resistivities of 5m and 50m at 12–22 km depth (Figure 7.4(a)).
START
STOP Generate or input a starting model (e.g., half-space).
Construct roughness matrix.
Compute misfit and roughness for current model.
Optimise misfit.
Is minimum misfit less than
required misfit?
Has model roughness
increased?
Compute step size and save model
Is (a) max. no. of iterations exceeded;
or (b) required misfit obtained, and, step size small/
model smooth?
Decrease step size.
Decrease stepsize.
Is minimum misfit less than minimum misfit from previous
iteration?
YES YES
YES
YES NO
NO
NO
NO Figure 7.2Flowchart showing
steps involved in least-structure inversion (modified from de Groot-Hedlin and Constable, 1990).
138 Inversion of MT data
Derive starting model from half-space by averaging -polarisation resistivities and performing 1-D inversion.
B
2-D inversion to fitB-polarisation phase data.
2-D inversion to fit -polarisation apparent resistivities and -polarisation phases.
E B
FixE-polarisation static shift factors.
2-D inversion to fit -polarisation apparent resistivities and phases.
E
2-D inversion to fit - and -polarisation apparent resistivities and phases.
E B
FixB-polarisation static shift factors.
Best-fit, least-structure 2-D model.
INPUT
Figure 7.3Flowchart showing successive introduction ofE-polarisation and B-polarisation apparent resistivities and impedance phases into the inversion process, following a scheme suggested by Smith and Booker (1991) for implementing RRI inversion, which incorporates a weighting criterion for estimating static-shift factors.
Depth (km)
0 12 22 50
100 5
500 (Ωm)
200 100 50 20 Period (s)
Phase ( )o
0 45 90
Period (s) 104 101
100 10–1 10–2
10–3 102 103 105
101 102 103
10 0
1 km
12 km
50Ωm100 m
22 km
410 km
4 km
(b)
(c) (a)
10 km
Horizontal distance (km) 104 101
100 10–1 10–2
10–3 102 103 105
100Ωm
500Ωm
300Ωm
5Ωm
Apparent resistivity (m)Ω
5mΩ 50mΩ
Resistivity
Figure 7.4(a) Layered model containing 10-km-thick (12–22 km) anisotropic layer with infinite lateral extension.
(b) Apparent resistivities and impedance phases generated by 2-D forward modelling of (a). (c) 2-D least-structure model obtained by inversion of twelve evenly spaced synthetic data sites, each having the anisotropic apparent resistivities and impedance phases shown in (b).
7.3 Artefacts of inversion 139
The individual dykes composing the anisotropic layer cannot be resolved since their respective lateral dimensions are much less than the depth to their tops.
The 2-D inversion (Figure 7.4(c)) yields high-conductivity blobs, with resistivities of between 5 and 10m, within larger blobs of between 10 and 20m within still larger blobs of between 20 and 50m. These blobs arise owing to the difficulty associated with fitting both polarisations of the apparent resistivities and impedance phases jointly.
Below the profile itself, we appear to have a basin-like structure that dips at an angle of208on either side, but we know that this again is purely an artefact of the inversion. There was no dipping structure in the original model.
The scenario in Figure 7.4 is one of the simplest examples of why we have to be careful about taking the results of 2-D inversion too literally. In this example, the original model had a fixed 2-Delectro- magnetic strike, so that E- andB-polarisations can be decoupled.
This is often not the case with measured data. Figure 7.5 shows a 2-D model generated by inverting data measured along a profile approximately perpendicular to the geological strike in NE England (Figure 5.12(b)). The modelled apparent resistivities and impedance phases of the two polarisations lie within two standard deviations of the measured data. The model shows a conductor (dark blob) of resistivity 10m extending downwards from depths of approxi- mately 12 km, in a background resistivity of 1000m. The dark blob has horizontal dimensions of 5–6 km, which at a depth of 10–12 km is below the theoretical resolution limit of the data. The blob is an artefact that arises only because the inversion scheme applied is required to fit divergent apparent resistivities and imped- ance phases (Figure 7.6) via a 2-D model. The divergence of the
0 40 30 20 10 0
2 4 6 8 10 12
Distance along profile (km)
Depth (km)
log10[ρ(Ω m)]
3.0 2.4 1.8 1.2 Figure 7.52-D least-structure
model generated by inverting data measured along a profile approximately perpendicular to the geological strike in NE England (see Figure 5.12(b)).
Arrows along top of model indicate site locations.
140 Inversion of MT data
differently polarised apparent resistivities and impedance phases probably arises owing to the imaging of the edge of a resistive granitic upper-crustal block, which bounds more conductive sedi- mentary basins, and owing to the proximity of the coastline – just 45 km away to the east, and 90 km away to the west. Although relatively shallow (50–70 m), the North Sea is underlain by thick (possibly > 5 km deep) sedimentary grabens. By incorporating the onshore and offshore conductivity structures into a 3-D model, and making reasonable assumptions about the conductances of the seawater and sedimentary basins, the observed impedance phase splitting can be reproduced, without the need for a conductive blob in the mid crust. In fact, a mid-crustal conductance of less than 200 S (that’s more than a factor of 10 less than in the 2-D blob model (Figure 7.5)) is sufficient to explain the data. Consideration of the lateral conductivity structures also explains the observation that the electromagnetic strike in NE England is period dependent (Figure 5.12(a)).
In our case study, using an average electromagnetic strike for the purpose of performing 2-D inversion of the MT data resulted in a model containing a conductive blob having horizontal dimensions that are not truly resolvable by the input data, and which on further investigation with 3-D forward modelling was found to be an artefact.
Models containing blobs occur frequently in electromagnetic litera- ture. The blobs apparent in these models arise primarily owing to an inability to fit both principal polarisations of the impedance tensor jointly using 2-D inversion. Anisotropy or three-dimensionality may give rise to this problem: be wary of models containing blobs at depth with lateral dimensions that are not resolvable by the input data!
90 45 0
104 103 102 101 100 104
103 102 101 100
90 45 0
10–310–210–1100 101 102 103 104 10–310–210–1100 101 102 103 104
10–310–210–1100 101 102 103 104 10–310–210–1100 101 102 103 104
Period (s) Period (s)
Phase (°) Phase (°)Resistivity (Ω m)
Resistivity (Ω m)
Figure 7.6Apparent resistivities and impedance phases for two MT sites along the profile from which Figure 7.5 is derived.
(Redrawn from Simpson and Warner, 1998.)
7.3 Artefacts of inversion 141