A comparison of smooth and blocky inversion methods in 2-D electrical imaging surveys

TLDR: In this paper, the L 2 norm based least squares optimisation method is used to map moderately complex structures with arbitrary resistivity distributions, and the blocky or L 1 norm optimization method can be used for such situations.

Abstract: Two-dimensional electrical imaging surveys are now widely used in engineering and environmental surveys to map moderately complex structures. In order to adequately resolve such structures with arbitrary resistivity distributions, the regularised least-squares optimisation method with a cell-based model is frequently used in the inversion of the electrical imaging data. The L 2 norm based least-squares optimisation method that attempts to minimise the sum of squares of the spatial changes in the model resistivity is often used. The resulting inversion model has a smooth variation in the resistivity values. In cases where the true subsurface resistivity consists of several regions that are approximately homogenous internally and separated by sharp boundaries, the result obtained by the smooth inversion method is not optimal. It tends to smear out the boundaries and give resistivity values that are too low or too high. The blocky or L 1 norm optimisation method can be used for such situations. This method attempts to minimise the sum of the absolute values of the spatial changes in the model resistivity. It tends to produce models with regions that are piecewise constant and separated by sharp boundaries. Results from tests of the smooth and blocky inversion methods with several synthetic and field data sets highlight the strengths and weaknesses of both methods. The smooth inversion method gives better results for areas where the subsurface resistivity changes in a gradual manner, while the blocky inversion method gives significantly better results where there are sharp boundaries. While fast computers and software have made the task of interpreting data from electrical imaging surveys much easier, it remains the responsibility of the interpreter to choose the appropriate tool for the task based on the available geological information.

Figures

Fig. 1. (a) A typical field arrangement for 2D electrical imaging surveys. (b) A cell-based model used for 2D resistivity inversion.

Fig. 2. (a) Wenner (alpha) array apparent resistivity pseudosection for a 100 Ω⋅m rectangular prism embedded in a 10 Ω⋅m medium. Models produced by the (b) smooth and (c) blocky inversion methods. The outline of the prism is shown in (b) and (c) for comparison.

Fig. 3. (a) A plot of the resistivity of the row of model cells between depths of 1.4 and 1.8 metres for the prism model with a sharp boundary of Figure 2. (b) A similar plot for the prism model with a boundary layer of Figure 6.

Fig. 4. (a) Wenner (alpha) array apparent resistivity pseudosection for a 100 Ω⋅m rectangular prism embedded in a 10 Ω⋅m medium with 3% random noise added. Models produced by the (b) smooth and (c) blocky inversion methods.

Fig. 5. (a) Wenner (alpha) array apparent resistivity pseudosection for a 100 Ω⋅m rectangular prism embedded in a 10 Ω⋅m medium with 10% random noise added. Models produced by the (b) smooth and (c) blocky inversion methods.

Fig. 6. (a) Apparent resistivity pseudosection for a 100 Ω⋅m rectangular prism embedded in a 10 Ω⋅m medium with a 33 Ω⋅m boundary layer. Models produced by the (b) smooth and (c) blocky inversion methods. The outline of the prism and boundary layer is shown in (b) and (c) for comparison.