1. GG 450 Feb 27, 2008 Resistivity 2

2. Resistivity: Quantitative Interpretation - Flat interface Recall the angles that the current will take as it hits an interface: Notice that where the current line goes through the interface it is bent towards a smaller angle, implying that  2 is greater than  1. Also note that deeper down the path goes through vertical.

3. Problem: Obtain formula that will yield the apparent resistivity for a model with a single interface separating two layers of different resistivity. We want to obtain the potential at arbitrary points P 1 and P 2, above above the interface and one below. This may seem like an impossible task, since the resistivity changes at the boundary, and energy will be both reflected from and transmitted through the boundary.

4. We can get a solution, though, by using the principle of IMAGE SOURCES. We reflect the actual source to the other side of the boundary, generating an image virtual source: We need the potential at P 1 for all possible ways that energy from the source can be seen there - as though the interface was a partly reflecting mirror.

5. We need to calculate the potential at P 1 from the paths r 1 and r 2, and r 3 for the potential at P 2. You can convince yourself that r 2 has the same length as the path to P 1 from the current source that is reflected at the interface. We do need to modify the potential where it reflects from or passes through the interface.

6. The energy reflected will be less than might be expected because some energy goes through the interface, with a reflection coefficient (k) and transmission coefficient (1-k). If the point where the potential is to be determined is across the boundary from the source, then the transmission coefficient will be used. Recall the potential: where D is the distance from the current electrode to the point of measurement. So, at P 1,

7. What’s the value of k? If both points P 1 and P 2 are on the interface, then the potentials should be equal, and: These formulas give us the basics for calculating the potential difference between two electrodes, and thus the apparent resistivity. Note that the model above doesn't include the earth's surface, which it must to be realistic. Note that above the earth’s surface,  1 is very large, so k 1,0 =-1. BUT…. now we have multiple reflections and image sources to worry about:

8. There are an infinite number of reflections that will contribute to the potential at P 1, but the energy in each successive reflection is less because energy is leaking into the lower layer, assuming that k 1,2 is not 1 (  2 very large).

9. From the above we note that the first reflection from the upper surface contributes to the potential by: All other reflections will involve reflections from both the top and bottom boundaries, and the squares and higher powers of the reflection coefficients. For the Wenner array: We can use this equation to calculate the potentials at the surface for various cases and generate models of what me might expect as the electrode spacings are changed. See resist.m. (Wenner array ONLY.)

10. In the figures below, the electrode configuration is the Wenner Array, where the geometry of the current and potential electrodes is constant, and the spacing as shown below: Measurements of apparent resistivity are taken at a range of values for a to obtain data to compare with model situations.

11. The above model shows a high resistivity below a low resistivity. Note that even though the depth of the layer is only 4 m, the complete picture requires the spacing BETWEEN electrodes to be over 300 m – that’s a total spread length of almost 1 km!

12. This is the opposite of the case above, with the higher resistivity on top. Now the spacing isn’t so bad, with total adequate spread length of only about 300 m.

13. What if the resistivity of the top layer changes without changing the depth to the 2nd layer? This is quite easy. The resistivity of both layers and the depth are easily recognized.

14. What if the resistivity of the deeper layer changes with a constant depth to the interface? Note that the depth of the interface is well constrained by the region where the resistivity first starts to change, but if the resistivity of the deeper layer is very high, spread lengths will need to be very long.

15. You usually don't need to go out to very large distances if all you care about is the depth to a layer of known resistivity. If all you care about is its depth, then it isn't necessary to go beyond the slope break, or a total array length of about 150m and 80m respectively in the models above. What if you have more than one resistivity interface below? In this case, the resistivity curve gets more complicated, but still fairly easy to interpret. Three cases for a three-layer model are shown below. The top one is for increasing resistivity with depth. Note that it looks VERY similar to the 2-layer case above.

16. You would have to be very careful to detect the intermediate layer!

17. Another model is shown below, with the medium resistivity at the top and the low at the bottom. Here, it's pretty obvious that there's something between the two layers.

18. The model below has the medium resistivity at the top and the high at the bottom:

19. Resistivity Uses

21. Profiling

22. Archeology

23. Groundwater

24. Pollution

25. MAGMA