Too complicated a start model.
Suppose the dimensionality analysis suggests a 3-D situation, and a 3-D code is selected. Even if the data indicate the existence of many different conductivity structures, one should not start with a com- plicated model. A better approach is to start with a simple model, and observe how the modelled transfer functions develop during the process of complicating the model.
126 Numerical forward modelling
Over-interpretation of transfer function amplitudes.
Because of the ‘distortion’ described in Chapter 5, the amplitudes of magnetotelluric transfer functions from field data can be poorly resolved. We therefore suggest to reproduce the impedance phase data with forward-modelling schemes. Geomagnetic transfer func- tions (induction vectors) can also be distorted by galvanic effects (Section 5.9), and this possibility should be considered when mod- elling these transfer functions with a code that simulates the induc- tion process. Theoretically, if large- and small-scale structures are included in the same model and if the model is large enough, the galvanic effects in the modelled transfer functions can be properly described.
MT data constrain conductances rather than resistivities and thicknesses of conductors.
Discriminating between a thick, moderately conductive anomaly and a thin, highly conductive anomaly is possible only with high- quality MT data acquired in regions of low geological noise.
Therefore, the thickness of a conductor is generally poorly con- strained. The constraints (or lack thereof ) on the thickness of a modelled conductivity anomaly can easily be investigated using forward modelling, by redistributing the conductance of a modelled anomaly into a thicker or thinner zone, and comparing the misfits that result from the different models. The proximity of apparent resistivity and impedance phase curves for different distributions (variable resistivities and thicknesses) of a 1000 S mid-crustal con- ductor is illustrated for a 1-D model in Figure 6.4 and for a 2-D
0 45 90
Period (s) 101
102
Period (s)
10–2 100 101 102 103 104
103 Resistivity (Ωm)
100 101 102 103
102 101 100 10–1
Phase ()ο Apparent resisitivity (m)Ω Depth (km)
10–1
10–2 10–1 100 101 102 103 104
Figure 6.4Comparison of apparent resistivity and impedance phase curves for three 1-D models in which a 1000 S mid-crustal
conductance is distributed in layers of 10 km10m, 5 km5m and 1 km1m, respectively.
The largest impedance phase difference of (68) for a 1-km-thick, 1m conductor versus a 10-km-thick, 10m conductor occurs at 4.1 s.
6.4 Avoiding common mistakes 127
model in Figure 6.5. The problem of distinguishing between thick and thin conductors is exacerbated if the overburden conductance is increased or the conductance of the anomaly is decreased.
The generalised model space might be too small.
This can lead to problems when attempting to model impedance phases (and apparent resistivities) and induction arrows jointly.
Induction arrows are often influenced significantly by regional conductivity anomalies that lie outside the MT model space. These different data sensitivities can be manifest as a lack of orthogon- ality betweenphase-sensitive strikeand induction arrow orientation.
The discrepancy can arise owing to current channelling, and can be modelled if the model space is sufficiently large and sufficiently finely gridded.
Being hooked on a band-wagon.
We should be aware of the possibility that we do have prejudices (although being open-minded people, otherwise). Here is an example:
we originally assume that conductivity is isotropic and try to inter- pret a 2-D dataset with a 2-D model. To reproduce the measured impedance phase data from both polarisations, we run the 2-D code twice – once for theE-polarisationand once for the B-polarisation (remember from Section 2.6 that the equations for the two polarisa- tions are decoupled and therefore they can be treated independently).
The two modes produce models that differ significantly in places, and attempts to model both modes jointly result in models with larger misfits. A possible solution could be to allow for anisotropic conductivity in those parts of the model that differ. The lesson here is that a large misfit is not always a problem, but can be a challenge. If we rise to this challenge, then we may re-think the underlying assump- tions in our model, rather than introducing or removing a few con- ductive blobs here and there. Of course, the opposite effect can also occur. For example, we are so much taken away with the concept of anisotropy that we interpret impedance phase differences between the two polarisations always with anisotropic models. The possibility that other features in the model are responsible for impedance phase differences must not be neglected!
128 Numerical forward modelling
400m x 100Ωm 500Ωm
250Ωm 10 km
1 km x 1 m 10 km x 10
Ω OR
Ωm 50 km
0 45 90
Phase ( )o
Distance along profile (km)
0 10 20 30 40 50 60 70 80
1000 S
10–2 10–1 100 102 103 104 0
45 90
Period (s)
Phase ( )o
*
(a) (b)
E-pol. B-pol. 1 km x 1 m E-pol. B-pol. 10 km x 10 m
Ω Ω E-pol. B-pol. 1 km x 1 m
E-pol. B-pol. 10 km x 10 m ΩΩ
101
Figure 6.5(a) Comparison of impedance phases at a period of 4.1 s along a profile that crosses a 52.5-km-wide, mid-crustal conductor having a conductance of 1000 S distributed uniformly within either a 1-km-thick block (closed symbols) or a 10-km-thick block (open symbols). (b) Comparison of impedance phases as a function of period generated by the two conductivity–thickness distributions in (a), for a site (*) located over the mid-crustal block.
Chapter 7
Inversion of MT data
Inversion schemes provide a ‘fast-track’ means for modelling data without the grind of forward modelling by allowing you to go from the data to the model, rather than the other way round. But beware – there are pitfalls, especially for the inexperienced!
In some inversion schemes the non-linear relationship between the model and the data is forced to be ‘quasi-linear’ by use of a transformation into a substitute model and data. In others, small model changes are assumed to have a linear relationship to small data changes. Once the required linear system of equations has been derived, the problem is reduced to one of inverting a matrix. The ‘Monte-Carlo inversions’ form another class of modelling algorithm. From a mathematical point of view, they are not strictly inversions at all, but rather long sequences of forward models in which conductivities are perturbed at random – as in a casino (a` la Monte Carlo).
Because electromagnetic energy propagates diffusively, MT sounding resolves conductivity gradients, rather than sharp boundaries or thin layers.
We should, therefore, think of MT as producing a blurred rather than a focussed image of the real Earth structure. Furthermore, because MT sound- ing is a volume sounding, the same features will be sampled by MT data at neighbouring sites when the inductive scale length is of the same order as or greater than the site spacing. Many inversion schemes are, therefore, founded on the philosophy of minimising model complexity, wherein, rather than fitting the experimental data as well as possible (which maximises the rough- ness of the model), the smoothest model that fits the data to within an accepted tolerance threshold is sought. On the one hand, ‘smooth’ or
‘least-structure’ models reduce the temptation to over-interpret data. On 130
the other hand, sharp boundaries do occur quite often in nature, an example being the contact between an ore body and its resistive host rock.
Although a good inversion scheme can provide the model with the smallest misfit, you should have some experience with forward modelling before you apply an inverse scheme: suppose you do a 1-D (layered half- space) inversion and find a conductive layer at some depth, which you excitedly proceed to interpret. A quick check with a 1-D forward scheme might have shown you that this layer is the product of a single data point. In other words, forward modelling is a good tool to explore the constraints provided by your data.
Many practitioners are content to compute one best-fit model for their data. Always be aware of the inherent non-uniqueness associated with model- ling, and explore more than one model. Forward modelling is a good way of exploring allowable perturbations to a model output from an inversion. What happens, for example, if your target area includes an electrically anisotropic domain, but your inverse scheme doesn’t allow for that possibility? Ironically, a lot of support for the anisotropy concept came from applications of 2-D inversions: applying the inversion to the transfer functions derived from electric fields in two perpendicular directions often produces two contradictory models, both of which have small misfit measures, indicating very different conductivities in a particular depth range. Again beware! Anisotropy is cer- tainly the easiest way to fit MT data for which the apparent resistivities and phases of the two principal polarisations differ, but there must be a full justification for preferring anisotropy to lateral effects.
Unfortunately, many users apply inverse schemes only because of their availability, in which cases even if the tests mentioned in Chapter 5 suggest a 3-D situation, a 2-D inversion is used because a 3-D inversion scheme is not available. On the other hand, 3-D modelling is computationally viable using forward-modelling algorithms.