Environmental and Exploration Geophysics II

1. Environmental and Exploration Geophysics II Gravity Methods (V) tom.h.wilson tom.wilson@mail.wvu.edu Department of Geology and Geography West Virginia University Morgantown, WV Tom Wilson, Department of Geology and Geography

2. Simple Geometrical Objects The question of edge effects addressed above takes advantage of simple geometrical objects, the plate and half plate, to answer questions about possible anomaly magnitudes associated with the problem at hand. See Stewart et al., section 6.5 pages 378 to 397 Tom Wilson, Department of Geology and Geography

3. Simple Geometrical Objects We make simplifying assumptions about the geometry of complex objects such as dikes, sills, faulted layers, mine shafts, cavities, caves, culminations and anticline/syncline structures by approximating their shape using simple geometrical objects - such as horizontal and vertical cylinders, the infinite sheet, the sphere, etc. to estimate the scale of an anomaly we might be looking for or to estimate maximum depth, density contrast, fault offset, etc. without the aid of a computer. Berger gets into a lot of the details in Section 6.5 of the text Tom Wilson, Department of Geology and Geography

4. Review of earlier discussions Let’s start with one of the simplest of geometrical objects - the sphere - Go to 31 Tom Wilson, Department of Geology and Geography

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8. What is the vertical component? Tom Wilson, Department of Geology and Geography

9. Hence …. Tom Wilson, Department of Geology and Geography

10. g directly over the center of the sphere is gmax z Tom Wilson, Department of Geology and Geography

11. gmax Tom Wilson, Department of Geology and Geography

12. Divide through by gmax gmax contains information about volume, density and radius Tom Wilson, Department of Geology and Geography

13. The shape of the curve gv/gmax is scale independent. It is not affected by the depth or size of the sphere. Tom Wilson, Department of Geology and Geography

14. Shape of the anomaly is independent of the size of the sphere that produced it. The shape, the variation as a function of x/z is the same for all spheres regardless of their depth or size. Tom Wilson, Department of Geology and Geography

15. At what point does the anomaly fall off to one-half of its maximum value? Tom Wilson, Department of Geology and Geography

16. Let the ratio g/gmax = ½ and solve for X/Z Tom Wilson, Department of Geology and Geography

17. X1/2 /Z = 0.766 implies that Z can be expressed in terms of X1/2 Tom Wilson, Department of Geology and Geography

18. X measures distance from the anomaly peak, and is NOT an absolute reference along the profile line. The location of the peak corresponds to X=0 Tom Wilson, Department of Geology and Geography

19. The “diagnostic position” is a reference location. It refers to the X location of points where the anomaly has fallen to a certain fraction of its maximum value, for example, 3/4 or 1/2. Tom Wilson, Department of Geology and Geography

20. In the above, the “diagnostic position” is X1/2, or the X location where the anomaly falls to 1/2 of its maximum value. The value 1.31 is referred to as the “depth index multiplier.” This is the value that you multiply the reference distance X1/2 by to obtain an estimate of the depth Z. Tom Wilson, Department of Geology and Geography

21. A table of diagnostic positions and depth index multipliers for the Sphere (see your handout). Note that regardless of which diagnostic position you use, you should get the same value of Z. Each depth index multiplier converts a specific reference X location distance to depth – to Z. Note that these constants (e.g. 0.02793) assume that depths and radii are in the specified units (feet or meters), and that density is always in gm/cm3. Tom Wilson, Department of Geology and Geography

22. What is Z if you are given X1/3? … Z = 0.96X1/3 In general you will get as many estimates of Z as you have diagnostic positions. This allows you to estimate Z as a statistical average of several values. We can make 5 separate estimates of Z given the diagnostic position in the above table. Tom Wilson, Department of Geology and Geography

23. You could measure off the values of the depth index multipliers yourself from this plot of the normalized curve that describes the shape of the gravity anomaly associated with a sphere. 2.17 See your handout 1.79 1.305 0.96 0.81 Tom Wilson, Department of Geology and Geography

24. Given X1/2 what is Z? Tom Wilson, Department of Geology and Geography

25. Refer to text for details … The Cylinder Tom Wilson, Department of Geology and Geography

26. Some details on the following slides … Tom Wilson, Department of Geology and Geography

27. Results for Horizontal Cylinder and Tom Wilson, Department of Geology and Geography

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29. We can ask the same kinds of questions we asked regarding the sphere. For example, Where does This tells us that the anomaly falls to ½ its maximum value at a distance from the anomaly peak equal to the depth to the center of the horizontal cylinder Tom Wilson, Department of Geology and Geography

30. Horizontal Cylinder Just as was the case for the sphere, objects which have a cylindrical distribution of density contrast all produce variations in gravitational acceleration that are identical in shape and differ only in magnitude and spatial extent. When these curves are normalized and plotted as a function of X/Z they all have the same shape. It is that attribute of the cylinder and the sphere which allows us to determine their depth and speculate about the other parameters such as their density contrast and radius. Tom Wilson, Department of Geology and Geography

31. How would you determine the depth index multipliers from this graph? 1.72 1.41 1 0.7 0.57 Tom Wilson, Department of Geology and Geography

32. X2/3 X3/4 X1/2 X1/3 X1/4 0.57 Z=X1/2 0.7 0.57 0.7 Locate the points along the X/Z Axis where the normalized curve falls to diagnostic values - 1/4, 1/2, etc. The depth index multiplier is just the reciprocal of the value at X/Z. X times the depth index multiplier yields Z Tom Wilson, Department of Geology and Geography

33. Again, note that these constants (i.e. 0.02793) assume that depths and radii are in the specified units (feet or meters), and that density is always in gm/cm3. Tom Wilson, Department of Geology and Geography

34. We should note again, that the depths we derive assuming these simple geometrical objects are maximum depths to the centers of these objects - cylinder or sphere. Other configurations of density could produce such anomalies. This is the essence of the limitation we refer to as non-uniqueness. Our assumptions about the actual configuration of the object producing the anomaly are only as good as our geology. That maximum depth is a depth beneath which an anomaly of given wavelength cannot have a physical origin. Maximum Depth Nettleton, 1971 Tom Wilson, Department of Geology and Geography

35. Which estimate of Z seems to be more reliable? Compute the range. You could also compare standard deviations. Which model - sphere or cylinder - yields the smaller range or standard deviation? Tom Wilson, Department of Geology and Geography

36. To determine the radius of this object, we can use the formulas we developed earlier. For example, if we found that the anomaly was best explained by a spherical distribution of density contrast, then we could use the following formulas which have been modified to yield answer’s in kilofeet, where - Z is in kilofeet, and  is in gm/cm3. Tom Wilson, Department of Geology and Geography

37. In-class group problems We will spend more time on simple geometrical objects during the next lecture, but for now let’s spend a few moments and review some additional problems from the text Pb-1:What is the radius of the smallest equidimensional void (such as a chamber in a cave - think of it more simply as an isolated spherical void) that can be detected by a gravity survey for which the Bouguer gravity values have an accuracy of 0.05 mG? Assume the voids are in limestone and are air-filled (i.e. density contrast = 2.7gm/cm3) and that void centers are never closer to the surface than 100. Tom Wilson, Department of Geology and Geography

38. Begin by recalling the list of formula we developed for the sphere. Tom Wilson, Department of Geology and Geography

39. Just for general discussion > (see 6.8, Burger et al.): The curve in the following diagram represents a traverse across the center of a roughly equidimensional ore body. The anomaly due to the ore body is obscured by a strong regional anomaly. Remove the regional anomaly and then evaluate the anomaly due to the ore body (i.e. estimate it’s deptj and approximate radius) given that the object has a relative density contrast of 0.75g/cm3 Tom Wilson, Department of Geology and Geography

40. residual Regional You could plot the data on a sheet of graph paper. Draw a line through the end points (regional trend) and measure the difference between the actual observation and the regional (the residual). You could use EXCEL or PSIPlot to fit a line to the two end points and compute the difference between the fitted line (regional) and the observations. Tom Wilson, Department of Geology and Geography

41. A. C. B. In a problem similar to problem 6.9 (Burger et al.) you’re given three anomalies. These anomalies are assumed to be associated with three buried spheres. Determine their depths using the diagnostic positions and depth index multipliers we discussed in class today. Carefully consider where the anomaly drops to one-half of its maximum value. Assume a minimum value of 0. Tom Wilson, Department of Geology and Geography

42. Due Dates • Remember that paper summaries and gravity labs are due Thursday, November 15th • Problems 6.5 and 6.9 are due Tuesday, November 13th. • Begin reading Chapter 7 on Exploration using Magnetic Methods … • We will introduce magnetic methods during the week of November 13th. Tom Wilson, Department of Geology and Geography