W is a diagonal

Fig. 7. Comparison of TEM inversion result and borehole data for station PF. The symbols are as for Fig. 6. Downhole Ž . logging Hawkins and Chadha, 1990; Fig. 5 suggested that the upper part of SS aquifer contains mudstone bands, hence the four-layer approach in TEM inversion. where the matrix E s W T

W, W is a diagonal

weighting matrix containing the reciprocals of Ž . 4 the observed data errors s , the quantity d i c 4 s W y q WAm is a kind of data, D is a Fig. 8. Comparison of TEM inversion result and borehole data for station CP. The symbols are as for Fig. 6. Fig. 9. Comparison of TEM inversion result and borehole data for station GH. The TEM sounding employed a 20-m-sided Tx loop. The symbols are as for Fig. 6. derivative regularisation matrix, B s b T b is a w matrix of undetermined multipliers, y s d y Ž .x f m is the discrepancy vector and A s Ž . E f m rEm is the matrix of partial derivatives Fig. 10. Comparison of TEM inversion result and borehole data for station BH. The symbols are as for Fig. 6. evaluated at an initial model, m . The field Ž . responses apparent resistivities are contained in d and those predicted by forward modelling Ž . are contained in f m . The vector h contains Ž the a priori parameter estimates inferred stratal . boundaries and resistivities towards which we wish to bias our solution. The above algorithm is applied in an iterative fashion. The logarithms of the TEM responses are considered in the Ž .4 inversion scheme, i.e. y s ln d y ln f m Ž .4 and A s E ln f m rEm as in standard prac- Ž . tice e.g. Meju, 1992, 1996 . The components of m are also taken to be the logarithms of the resistivities and interface depths of the sought subsurface model. The above algorithm can be interpreted as incorporating some kind of a priori constraints in the inversion process as a partial remedy to the problem of non-unique- Ž . ness in inversion cf. Jackson, 1979 especially since we seek only a particular class of interpre- tive resistivity models; an illustrative example of this model construction approach is shown in Fig. 4. Model bounds are estimated using the non-linear most-squares technique described in Ž . Meju 1994 ; the solution envelopes were com- puted for a threshold misfit set equal to 1.1 Ž 2 . times the least-squares x misfit for each Ž . station see Meju and Hutton, 1992 as in the example shown in Fig. 5 for the optimal model presented above in Fig. 4. The TEM data from all the borehole sites have been inverted using the approach discussed above. The resulting models are shown in Figs. 6–11. In each figure, the optimal least-squares model and the initial transformation result are shown together with the depth locations of the main lithological boundaries from Table 1. The interpretative most-squares model bounds for the least-squares models are given in Table 2. For convenience, these models will be referred to as the biased estimation models in this paper. All the TEM data were also inverted using a Fig. 11. Comparison of TEM inversion result and borehole data for station GS. The symbols are as for Fig. 6. Note the Ž . resistivity undershoot depressed apparent resistivity curve around 0.07–0.2 ms whose effect might be misinterpreted as a separate conductive layer around 20 m depth on the simple resistivity–depth transformation. Table 2 Summary of parameter ranges from constrained most-squares inversion for all the TEM stations. The interpreted correlation Ž . with lithology Table 1 is shown in the last column. The question marks denote unequivocal interpretations Ž . Ž . Stationrlayer number Resistivity V m Depth to base m Depth-correlated lithology LH 1 28–30 9–10 Sandrupper boulder clay ? 2 23–25 22–26 Lower boulder clay 3 34–38 43–48 SS with marl bands 4 59–72 SS PF 1 22–25 8–9 Sandrupper clay drift ? 2 18–20 25–28 Lower clay drift Ž . 3 22–26 41–46 SS muddy section ? 4 30–37 SS CP 1 28–37 9–11 Sandrclay drift ? 2 13–16 28–32 Mudstone Ž . 3 26–30 43–50 SS muddy section ? 4 42–65 SS GH 1 17–22 7–8 Sandrupper clayey drift ? 2 14–16 21–24 Lower clay drift 3 56–64 67–74 Gypsumranhydrite mudstone 4 21–30 SS BH 1 21–22 7.8–9 Sandy clay drift ? 2 19–20 45–50 Mudstone Ž . 3 40–48 110–125 Marl and mudstone gypsiferous ? 4 29–37 SS GS 1 10.5–12 7.5–8.1 Boulder clay ? 2 17–20 45–50 Marl and mudstone 3 105–141 170–192 Gypsiferousranhydritic marl 4 4.8–5.2 SS ? common interpretational approach in which the layer boundaries correspond to those stratal Ž boundaries indicated in the borehole logs Table . 1 and are held fixed while the resistivity pa- rameters are free to vary. The resulting models will be called fixed-depth models and are also presented in Figs. 6–11 for comparison; the resistivity deduced for the various lithological units from this alternative TEM inversion are given in Table 1. Except for station GH, only the inversion results for soundings with 50-m- sided Tx loops will be presented in this paper.

4. Geological appraisal of TEM models