structure philosophy, or a treatise on the difference between the mathematical and the physical
viewpoint
Many inversion schemes are founded on linearisation about an arbitrary starting model of an inherently non-linear system of equa- tions, as described in the previous section. However, Parker (1980) presented a non-linear formulation whereby the optimal 1-D solu- tion is described by a stack of delta functions (infinitesimally thin sheets of finite conductance) embedded in a perfectly insulating
7.2D+optimisation versus least-structure modelling 133
half-space. This is known as theD+model. The fit of the modelled D+ data to the measured data can be expressed in terms of the weighted least-square statistic given in Equation (6.6), withjsites=1.
It should be noted that this"2misfit statistic depends, in part, on the distribution of errors in the measured data. Because of measurement errors and inevitable departures from the theoretically idealised 1-D dimensional sounding environment,"2is not expected to vanish to zero. The expected value of"2for data with independent Gaussian errors isifreq. If models whose misfits are less than two standard deviations of "2 above its expected value are deemed acceptable, then this defines a misfit inequality (cf. Parker and Whaler, 1981):
"25ifreqþ2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ðifreqÞ
p : (7:3)
Failure to satisfy this inequality implies that data cannot be satis- factorily modelled following the assumption of a 1-D Earth, and that multi-dimensional conductivity structures must be considered.
In limited situations, the calculation ofD+solutions may represent a useful initial stage to modelling, offering confirmation, or otherwise, as to the existence of 1-D solutions. However, such sharp interfaces as delta functions are physically unrealistic. Parker and Whaler (1981), therefore, developed theH+model, in which the best-fit to a model composed of a stack of homogeneous layers overlying a perfect con- ductor placed at the maximumpenetration depthis sought. However, depending on the error structure of measured data, approaches to modelling aimed solely at minimising the misfit between modelled and measured responses are not necessarily justified. Fitting electromag- netic data as closely as possible maximises theroughnessof a model, and it is generally desirable to sacrifice a degree of fit to a dataset in favour of introducing gradational conductivity contrasts (Constable et al., 1987).
The conductivity structure apparent in models derived from layered 1-D inversion schemes can depend quite critically on an often arbitrarily pre-determined number of layers that should be present in the final model. Under-parameterisation of the inverse problem tends to suppress structure that could be significant, whilst over-parameterisation introduces structure that is redundant, and not truly resolvable by the available data (Constableet al., 1987).
Also, models with too many layers may develop oscillations as the modelled structure tends to the delta function (D+) solution that optimises the misfit. One solution to this problem is the minimum layer inversion (Fischeret al.1981), in which additional layers are introduced into a model at progressively greater depths only when successively longer-period data demands their presence. On the
134 Inversion of MT data
other hand, Constable et al. (1987), drawing on the philosophy embodied in the tenet known as Occam’s razor10, propose finding the smoothest possible model (known as theleast-structure model) consistent with an acceptable (user-definable) fit to the data. Given the diffusive nature of electromagnetic fields, passive EM tech- niques resolve conductivity gradients rather than sharp bound- aries at depth, giving least-structure models a certain appeal. By explicitly minimising structural discontinuities, unjustifiably com- plex interpretations of data are avoided, and any structure present within the model should lie within the resolving power of the data.
Therefore, the temptation of over-interpreting data is reduced.
However, as we shall see in Chapters 8 and 9, in certain depth ranges such as at the lithosphere–asthenosphere boundary and at the 410 km transition zone, the existence of discontinuities is widely expected. Therefore, layered models and least-structure (smooth) models represent two extremes between which the resistivity–depth distribution of the real Earth is likely to lie.
2-D models are constructed on a rectangular grid of rows and columns intersecting at nodes in they–zco-ordinate plane (as, for example, delineated in Figure 2.5) to form blocks or cells, each of which is attributed a uniform conductivity. Rules of thumb for constructing such a mesh are dealt with in Section 6.2. Once defined, the mesh remains fixed from one iteration to the next. Most 2-D inversion algorithms (e.g.,Occam Inversion(de Groot-Hedlin and Constable, 1990);Rapid Relaxation Inversion(RRI) (Smith and Booker, 1991)) are founded on the least-structure philosophy, and involve joint minimisation of data misfit (e.g., expressed as an rms statistic as given in Equation (6.6)) and model roughness. For 1-D models, roughness can be defined as
MR1¼
ð dm zð Þ dz
2
dz (7:4)
or
MR2¼
ð d2m zð Þ dz2
2
dz (7:5)
wheremðzÞis either conductivity,, orlnðÞ(Constableet al., 1987).
In 2-D models, both horizontal and vertical conductivity gradients can be minimised (e.g., by minimising conductivity differences
10Named after William of Ockham, a fourteenth-century Franciscan monk who wrote ‘Plutias non est ponenda sine necessitate’. (Entities should not be multiplied unnecessarily.)
7.2D+optimisation versus least-structure modelling 135
between laterally and vertically adjacent cells in the model). This process is sometimes known as ‘regularisation’, and the grid on which it is performed is called the regularisation mesh(de Groot- Hedlin and Constable, 1990). For example, in this case, we can write the roughness function, QðyiÞ, as a scaled norm of the Laplacian beneath siteyi(Smith and Booker, 1991):
Q yð Þ ¼i
ð d2m yð i;zÞ
df zð Þ2 þg zð Þd2m yð i;zÞ dy2
y¼y
i
d2z df zð Þ2
!2
df zð Þ; (7:6)
wheregðzÞallows for a trade-off between penalising horizontal and vertical structures, andfðzÞcontrols the scale length for measuring structure at different depths. Choosingm¼lnð Þ andf¼lnðzþz0Þ generally produces misfits between modelled and measured data that are uniform across the frequency spectrum (Smith and Booker, 1988). The non-dimensional weight function gðzÞ can be generalised as (Smith and Booker, 1991):
gðzÞ ¼ i zþz0
; (7:7)
whereiis the distance between measurement sites neighbouringyi, and and are constants (that are unrelated to phase-sensitive strike and skew). As i! 1, g zð Þ ! 1. Therefore, horizontal gradients are penalised more relative to vertical gradients the greater the site spacing. The constantscales the horizontal versus vertical derivatives in Equation (7.7). The larger the value of , the greater the degree of horizontal smoothing. Small values of favour shallow structure, but as is increased from 0 to 1.5, the penalty for horizontal structure becomes independent of depth.
Default values of ¼4 and ¼1:5 are suggested by Smith and Booker (1991).
The roughness operator and misfit are jointly minimised via the operatorWi:
Wið Þ ¼yi Q yð Þ þi ie2i; (7:8)
where i is a trade-off parameter between model structure and misfit, andP
e2i is the standard2statistic. Since the inverse problem is non-linear,2is reduced in small steps over a number of iterations so that assumptions inherent in the linearisation (e.g., Equation (7.2)) are not violated.
Changes in the model structure that are required by data at (an)other site(s) can produce a local increase inWi. This is particu- larly true for B-polarisation data, which are sensitive to electric
136 Inversion of MT data
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).