DCIP2D:
Inversion of 2D IP data

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To invert the IP data in Fig 4b we first linearize equation (5). Let _i and _i denote the chargeability and electrical conductivity of the i^th cell. Linearizing the potential about the conductivity model yields

      (11)

Substituting into equation (5) yields

      (12)

This can be approximately written as

     (13)

Thus the i^th datum is

      (14)

where

      (15)

      (16)

where _d^* is a target misfit. In practise the true conductivity is not known and so we must use the conductivity found from the inversion of the DC resistivity data to construct the sensitivity matrix elements in equation (15).

The functional in equation (16) can be minimized directly but we need to ensure that the recovered chargeability is positive. In the inversion of the DC potentials to recover the conductivity we ensured positivity by working with ln as the model in the inversion and applying the model norm to that quantity. That was natural since conductivity varies over many orders of magnitude and it is the variation of conductivity that is diagnostic of earth structure. Intrinsic chargeability is confined to the region [0,1). Moreover, we are not generally interested in the variation of chargeability in the range between zero and some small number (e.g. 0.01). Working with logarithmic values however, puts undue emphasis on these small values. An efficient method by which to solve the linear inverse problem with positivity constraints is through a non-linear mapping of variables. The details of the IP inversion algorithm can be found in Oldenburg and Li (1994).

The 124 IP data shown in Fig 4b will now be inverted. The model norm objective function is identical to that used for the DC resistivity inversion except that the reference model has been set equal to zero. A zero reference model is likely to be appropriate for most IP inversions. The resultant chargeability model which has the desired misfit of 124 and was obtained after 11 iterations, is shown in Fig 6b. There has been an excellent recovery of both the chargeable surface layer and the subsurface blocks. Some roughness in the surface block is noticed but the amplitude range of .045 to 0.07 brackets the true value of 0.05. The locations of the chargeable blocks have been well reproduced and the maximum amplitude of the model is consistent with the true value. However, the centre of the chargeable block on the right is slightly deeper than the true position, and the smooth bottom of the recovered anomaly has accentuated this appearance. Quantification of the benefits of the inversion can be obtained by comparing Fig 6b with the data in Fig 4b. We also note that this good recovery was obtained even though the sensitivities for the IP inversion were generated from the recovered conductivity in Fig 5b rather than the true conductivity in Fig 5a.

Appendix: If there is a need to incorporate a known dip into the inversion, see Li, Y., and Oldenburg, D.W. (2000). There are instructions for incorporating dip with DCIP2D in a short appendix (PDF) to the manual.