6.2.1 Gridding rules and boundary conditions
We refer the reader to Weaver (1994) for a summary of work done on 2-D modelling. With respect to 3-D modelling, differential equation
6.2 Gridding rules, boundary conditions, and misfits 119
methods and integral equation methods have been proposed.
Whereas in differential equation methods both the conductivity anomaly and the surrounding ‘normal world’ is covered with a net- work of cells, in the integral equation technique only the anomalous conductivity structure is ‘gridded’. The latter requires less computa- tional power and it is probably for this reason that most of the early contributions were devoted to the integral equation method (e.g., Raiche, 1974; Hohmann, 1975; Weidelt, 1975; Wannamakeret al., 1984b). Considering only the region with anomalous conductivity is made possible by a volume integration of Maxwell’s equations in conjunction with Green’s theorems (Zhdanovet al., 1997). The elec- tromagnetic field in the model is described by the Fredholm integral equation:
UðrÞ ¼UnðrÞ þ ð
Va
Gðr=r0Þjðr0ÞdV0 (6:1)
for the vector fieldU(either the electric or the magnetic field). Here, G is Green’s tensor, j¼aE is the current density owing to the anomalous conductivity, a, and Va is the volume of the region with anomalous conductivity. For the numerical solution, Va is subdivided into cells in which the functions U(r) and aðrÞ are assumed to be constant. Hence, the problem is reduced to the solution of a linear system of equations, in which the integral in Equation (6.1) is replaced by a sum.
Integral equation methods have rarely been used for interpret- ing measured electromagnetic data, probably because the model of an isolated ‘anomaly’ surrounded by a layeredhalf-spacegenerally does not describe realistic tectonic situations. In contrast, differen- tial equation methods allow for the computation of electric and magnetic fields in arbitrarily complex conductivity structures, but require more computational power (both storage space and com- puting time). Using these methods, a boundary value problem is solved either with finite differences (FD) or with finite elements (FE).
With few exceptions (e.g., the FE code by Reddyet al., 1977) most applications of differential equation methods to EM rely on FD techniques. Here, the linear relations between the fields at the nodes of a 3-D grid are of interest. In the early FD techniques, differential equations – either the Maxwell equations (Equation (2.6)) or the induction equations (Equations (2.14) and (2.15)) – were replaced by finite differences. For example,
dU
dx ¼Uðx2;y;zÞ Uðx1;y;zÞ
x ; (6:2)
120 Numerical forward modelling
where Uis any field component and x is the distance between neighbouring nodes (x1, y, z) and (x2, y, z). For example, the y-component of Equation (2.6b):
dBx dz dBz
dx ¼0Ey is replaced by
Bxðx;y;z2Þ Bxðx;y;z1Þ
z Bzðx2;y;zÞ Bzðx1;y;zÞ
x ¼0Eyðx;y;zÞ:
(6:3)
Modern FD applications (Mackie et al., 1993; Mackie and Madden, 1993) use the integral form of Maxwell’s equations. For example, the integral form of Ampe`re’s Law:
þ Bdl¼
ð ð
Eds (6:4)
is translated into a difference equation (instead of Equation (2.6b)).
Similarly, the integral form of Faraday’s Law:
þ Edl¼
ðð
i!Bds (6:5)
is the integral formula equivalent to Equation (2.6a). The numerical realisation of discrete spatial differences (of which Equation (6.2) is the simplest example) is replaced by a numerical averaging that simulates the integrals in Equations (6.4) and (6.5).
Thin-sheet modelling takes advantage of the inherent ambi- guity involved in distinguishing between the conductivity and thickness of conductivity anomalies by squashing anomalous conductances into an infinitesimally thin layer (Figure 6.1). This approach reduces the computational problem from one of calcu- lating perturbations in two parameters – conductivity and thick- ness – to one of calculating perturbations in one parameter – conductance. For example, the thin-sheet modelling algorithm of Vasseur and Weidelt (1977) allows for a 3-D structure within one layer of an otherwise layered Earth. Thin-sheet modelling algo- rithms have been successfully applied in studies involving land–
ocean interfaces and seawater bathymetry when the targeted electromagnetic skin depth is greater than the seawater depth (see Section 9.4). Wang and Lilley (1999) developed an iterative approach to solving the 3-D thin-sheet inverse problem, and applied their scheme to a large-scale array of geomagnetic induction data from the Australian continent.
Squash
τ τ τ τ τ τ τ τ τ1 2 3 4 5 6 7 8 9
σ h
Figure 6.1Illustration of the concept of thin-sheet modelling: a conductivity anomaly with conductivity,, and variable thickness is represented within an infinitesimally thin sheet composed of discretised conductances (1,2,3, etc).
6.2 Gridding rules, boundary conditions, and misfits 121
An early application of 3-D FD modelling (Jones and Pascoe, 1972) by Lines and Jones (1973) illustrates how sign changes of components of the electromagnetic fields with periods of order 30 minutes might arise from island effects. More general applications of 3-D FD modelling techniques to tectonic and geodynamic prob- lems (e.g., Maseroet al., 1997; Simpson and Warner, 1998; Simpson, 2000, Leibecker et al., 2002) came later than thin-sheet model studies, one reason for this tardiness being the computational power required if all rules for the design of a 3-D grid are followed properly.
Weaver (1994) provides some practical rules for the design of a grid for 2-D FD forward modelling, which can, with some modifi- cations, also be applied to 3-D FD modelling.
‘The following guidelines. . .have been found helpful in designing a satisfactory grid
1 Up to at least two electromagnetic skin depths on either side of a vertical or horizontal boundary separating two regions of different conductivity, the node separation should be no more (and very close to the boundary preferably less) than one-quarter of an electromagnetic skin depth. At greater distances where the field gradient will be smaller, this condition can be relaxed.
2 The first and last (Mth) vertical grid lines should be placed at least three electromagnetic skin depths beyond the nearest vertical conductiv- ity boundary. Here ‘electromagnetic skin depth’ refers to the electro- magnetic skin depth in the most resistive layer of the relevant 1-D structure.
3 The separation of adjacent nodes should be kept as nearly equal as possible near conductivity boundaries, and preferably exactly equal across the boundaries themselves. When a rapid change in the node separation from one part of the model to another is required, the transi- tion should be made as smoothly as possible, preferably in a region where field gradients are expected to be small and with no more than a doubling (or halving) of adjacent separations.’
Computing model data for different periods will result in dif- ferent electromagnetic skin depths being realised. (‘Two electromag- netic skin depths’ is a larger distance at long periods than at short periods). Weaver’s guideline 1 implies that the node separation can be increased with increasing distance from a boundary. However, when we increase node separations, we must take care not to violate
122 Numerical forward modelling
guideline 3. In order to follow guidelines 1 and 3, we might be forced to choose small (1/4 of the smallest electromagnetic skin depth) node separations over a very wide area. Weaver’s guideline 2 is a practical consequence of the ‘adjustment length’ concept (Section 2.7): a vertical conductivity boundary can influence the electromag- netic field as far away as two to three electromagnetic skin depths.
This guideline has its equivalent for 3-D FD modelling. Horizontal and vertical cross-sections through a grid employed for 3-D FD modelling (Mackieet al., 1993) of an array of 64 MT sites stationed over a 300400 km2area in Germany are shown in Figure 6.2. In contrast, in integral equation methods, the grid only covers the anomalous domain (e.g., Raiche, 1974).
Designing a grid involves a trade-off between computational econ- omy and precision. A practical way to explore the effect of node separation and the model size on the modelled data could be the following: for the same model, we start by designing a grid where the above guidelines are observed to some extent; we subsequently decrease the node separation and increase the number of nodes. If the model data do not change, then our starting grid was already fine enough. On the other hand, a perturbation of the model data indicates that a finer grid is required, and the sequence should be repeated.
6.2.2 Misfits revisited
In Section 5.3, we introduced misfit measures describing the applic- ability of general models (e.g., the layered-Earth model or the 2-D/3-D superimposition model) to a particular dataset. More generally, a misfit is a difference between measured and modelled data, e.g.,:
"2¼Xjsites
j¼1
Xifreq
i¼1
CijCij;modj2
Cij
2 ; (6:6)
whereCij,Cij;modandCijare the measured and modelledtransfer functions at site j and evaluation frequency i and the confidence intervalof the measured data, respectively. Of course, for the case of 1-D models, only one site is evaluated.
Is it desirable to have a very small misfit? Too small a misfit can mean that the model resolves data noise: suppose we have 10 sites and 20 discretised frequencies, then the misfit given in Equation (6.6) should not be smaller than 200, because for a misfit smaller than 200 the average distance between measured and modelled data will be smaller then the confidence interval. However, practical experience tells us that there is rarely a model with such a small
6.2 Gridding rules, boundary conditions, and misfits 123
N
z
Figure 6.2Example of a grid used in 3-D FD modelling. The horizontal cross-section (top) consists of 103119 cells and represents a geographical area of 858923 km2in Germany. The black dots indicate the locations of MT sites. These lie within the core of the grid, where cell dimensions are 55 km2. The vertical cross-section (bottom) contains 34 layers that extend to a depth of 420 km. Boundary conditions are imposed by a 1-D background model. Overall, the grid should be envisaged as a cube with 10311934 cells. (Courtesy of A. Gatzemeier.)
124 Numerical forward modelling
misfit. For 1-D and 2-D modelling, this is so because the modelled data are rarely truly 1-D or 2-D: even if the dimensionality criteria discussed in Section 5.3 suggest a 3-D model, MT practitioners often use 2-D modelling.
Even if 3-D modelling is performed, the data will often not support a quest for the best-fit 3-D model, because the area repre- sented by the 3-D model is not covered densely enough with MT sites.
Is 3-D modelling useful at all in this situation? Yes, if we use forward modelling forhypothesis testing: by running different models, we can assess whether a particular feature in a model is reallyrequiredby a particular dataset – if so, the misfit will be lowered by inclusion of the feature in question. The trade-off between complicated and rough models, which generate a small misfit, and ‘smooth’ models, which generate larger misfits, will be discussed in Section 7.2.
The misfit parameter expressed in Equation (6.6) is an arbitrary mathematical quantity, and because it is possible to derive models that have the same misfits, but which contain more or less structure than is present in the data, it is worthwhile to compare the shapes of the modelled and measured transfer functions (see, for example, Figure 6.3).