1. Environmental and Exploration Geophysics I
Resistivity II
tom.h.wilson
Department of Geology and Geography
West Virginia University
Morgantown, WV

2. For next class write up the answer to in-class problem 2 and hand in next period.
Read over problems 1 through 3 in today’s handout and be prepared to ask questions about them this coming Tuesday.
I’ll summarize the approach to these problems in class today.

3. What kind of an array is this? What are d1, d2, d3 and d4 ?
Homework problem 1a
20m
source
sink
Surface   4m
Depth
12m 2m
=200-m   P1 P2
Find the potential difference between points 1 and 2.
What kind of an array is this? What are d1, d2, d3 and d4 ?

4. Use basic equations for the potential difference.
The critical point here is that you accurately represent the different distances between the current and potential electrodes in the array.

5. 2. Current refraction rules
Given these resistivity contrasts - how will current be deflected as it crosses the interface between layers? Measure the incidence angle and compute the angle of refraction.

7. What’s your guess?
2 > 1
tan  increases with increasing angle
2 < 1

8. 1 ?
2 > 1 2
2 varies as 1 and 1 varies as 2

9. ?
2 > 1

10. ?
2 < 1

11. ?
2 > 1

12. Let’s consider the in-class problem handed out to you last lecture.
Incorporating resistivity contrasts into the computation of potential differences.
3. Calculate the potential at P1 due to the current at C1 of 0.6 amperes. The material in this section view extends to infinity in all directions. The bold line represents an interface between mediums with resistivities of 1 and 2. m
1=200-m
1=50 -m C1 P1 
Let’s consider the in-class problem handed out to you last lecture.

13. In-Class/Takehome Problem 2
In the following diagram -
Suppose that the potential difference is measured with an electrode system for which one of the current electrodes and one of the potential electrodes are at infinity. Assume a current of 0.5 amperes, and compute the potential difference between the electrodes at
PA and . Given that d1 = 50m, d2 = 100m, 1 = 30-m, and 2 = 350-m.

14.  Current reflection and transmission Sink Source Electrode 1=30-m
One potential electrode d1 PC a PA PB b
2=350-m
d2 = a+b
Image point

15. At PA
1=30-m
Some current will be transmitted across this interface and a certain amount of current (k) will be reflected back into medium 1. d1 PA a ? PA PA b
2=350-m
d2 = a+b
Reflection point
Image point

16. Use of the image point makes it easy to estimate the length along the reflection path
Path length is distance from image point to PA.

17. Potential measured at A
k is the proportion of current reflected back into medium 1. k is also known as the reflection coefficient.

18. Potential measured at point B
1-k is the transmission coefficient or proportion of current incident on the interface that is transmitted into medium 2.

19. Potential measured at point C

22. Potentials a hair to the left or right of the interface should be approximately equal.

23. Locate an image electrode and incorporate reflection process,
Incorporating resistivity contrasts into the computation of potential differences.
3. Calculate the potential at P1 due to the current at C1 of 0.6 amperes. The material in this section view extends to infinity in all directions. The bold line represents an interface between mediums with resistivities of 1 and 2. m
1=200-m
1=50 -m C1 P1 
Locate an image electrode and incorporate reflection process,

24. General ideas about potential field and current distributions.

26. From the text we are given -
z is depth and d is electrode spacing

28. In the preceding diagram z was depth
In the preceding diagram z was depth. In this diagram z is depth divided by a which is 1/3rd the current electrode separation. Remember the a-spacing is referred to in the context of the Wenner Array.

29. Change in potential
d in this case is the distance between the source electrode and nearby potential electrode; i.e a in the case of the Wenner array.
Note the similarity of this “sensitivity” curve to the relative response function (V(z)) used with terrain conductivity data. You can also think of this curve as indicating the contribution of intervals at various depths to the potential between one current and one potential electrode.

30. We see that in general for the Wenner array the peak sensitivity of the array to subsurface resistivity distributions occurs at depths approximately equal to the a-spacing.

31. By comparison to the characterization of instrument response as a function of depth and intercoil spacing these relationships are defined much more qualitatively for the resistivity applications.

32. Qualitative interpretation of a resistivity sounding -
Go to board and discuss what a sounding is. What do you guess you are seeing here?
How many layers have been sensed in this resistivity sounding?
The observations consist of apparent resistivities recorded at various a-spacings (Wenner array) or l-spacings (Schlumberger array).

33. Let’s examine the utility of the interpolation approach using synthetic data. Synthetic data are data that have been calculated from a model. Thus we know what the actual answer should be. Refer to your handout.
Given the potential how do we determine apparent resistivity?

34. Where G = 2a for the Wenner array
The apparent resistivities (a) are computed from the relationship we derived earlier.
Where G = 2a for the Wenner array

35. Here are some plots of our synthetic or “test” data set
Here are some plots of our synthetic or “test” data set. The model from which it is derived is shown at lower right.

36. The Inflection Point Depth Estimation Procedure
This technique suggests that the depths to various boundaries are related to inflection points in the apparent resistivity measurements. Again, the In-Class data set illustrates the utility of this approach.
Apparent resistivities plotted below are shown over the model for both the Schlumberger and Wenner arrays. The inflection points are located, and dropped to the spacing-axis. The technique is suggested too be most applicable for use with the Schlumberger array. The inflection point rule varies with array type. For the Wenner array, the approximate depth to the interface is 1/2 the inflection point spacing. For the Schlumberger array this would give a depth to the top of the layer of about 12 meters instead of the actual depth of 8. In the case of the Wenner array we would get a depth of about 9 meters. A general “rule of thumb” for the Schlumberger array would to divide the inflection point distance by 3 instead of 2; that would yield 8 meters.

37. Resistivity determination through extrapolation
This technique suggests that the actual resistivity of a layer can be estimated by extrapolating the trend of apparent resistivity variations toward some asymptote, as shown in the figure below. The problem with this is being able to correctly guess where the plateau or asymptote actually is. Spacings in the In-Class data set only go out to 50 meters. The model data set (below) used for the inflection point discussion reveals that this asymptote is reached only gradually, in this case at distances of 500 meters and greater. Since most of the layers affecting the apparent resistivity in our surveys will be associated with thin layers, we are unlikely to be able to do this very accurately. The apparent resistivity will vary considerably over that distance rather than rise gradually to resistivities of individual layers. At best the technique offers only a crude estimate.

38. Method of Characteristic curves
The curves are calculated responses for given conditions. The curves shown at right are for a two-layer model.
We’ll consider this method in more detail during lecture next Thursday.

39. Equivalence - non-uniqueness ...

40. The realm of possibility ...

41. Multi-electrode resistivity systems

43. Tri-potential resistivity method

44. Normal Wenner array configuration

53. Some background information about the resistivity lab

57. Frohlich used the method of characteristic curves to estimate the depths to resistivity interfaces and their resistivity. We’ll talk more about the method of characteristic curves in the next lecture.

58. Next Class Hand in homework problems 1-3 next Thursday.
Skim over Frohlich’s paper before lab next Tuesday