1. Environmental and Exploration Geophysics I tom.h.wilson tom.wilson@mail.wvu.edu Department of Geology and Geography West Virginia University Morgantown, WV Magnetic Methods - III

2. Vertically polarized sphere or dipole Vertically polarized vertical cylinder Vertically polarized horizontal cylinder

3. Vertically polarized sphere or dipole Vertically polarized vertical cylinder Vertically polarized horizontal cylinder

4. These distances are referred to as diagnostic positions. Thus in the plot below, the points along the x axis where the anomaly falls off to 3/4 ths, 2/3 rds, 1/2, 1/3 rd and 1/4 th of the maximum value of the anomaly are referred to as X 3/4, X 2/3, X 1/2, X 1/3 and X 1/4, respectively. X 2/3 X 1/2 X 1/3 X 1/4

5. Those measurements provide us with the above table. In this case we have distances in multiples of x/z.

6. In working with an actual anomaly, we can measure the actual distances to points where a given anomaly drops to various fractions of the maximum anomaly value and then compute the depth z.

7. We measure the distances (x) to the various diagnostic positions and then convert those x’s to z’s using the depth index multipliers which are just the reciprocal of the x/z values at which the anomaly drops to various fractions of the total anomaly magnitude.

8. is a function of the unit-less variable x/z The vertical field is often used to make a quick estimate of the magnitude of an object. This is fairly accurate as long as i is 60 or greater Dipole/sphere Horizontal cylinder Vertical cylinder

9. X/Z Vertical Cylinder SphereHorizontal Cylinder X 3/4 0.460.3150.31 X 1/2 0.7660.50.495 X 1/4 1.230.730.68 Depth Index Multipliers Vertical Cylinder SphereHorizontal Cylinder X 3/4 2.173.183.23 X 1/2 1.30522.02 X 1/4 0.811.371.47

10. Cylinder or sphere Background noise at the site is ± 5nT Formula for the horizontal cylinder Formula for the sphere

11. Sphere vs. Vertical Cylinder; z = __________ Diagnostic positions Multipliers Sphere Z Sphere Multipliers CylinderZ Cylinder X 3/4 = X 1/2 = X 1/4 = The depth 0.9 1.55 2.45 2.86 3.1 3.35 1.95 2.03 2.00 2.17 1.31 0.81 3.18 2 1.37 diagnostic distance

12. Diagnostic positions Multipliers Sphere Z Sphere Multipliers Cylinder Z Cylinder X 3/4 3.182.17 X 1/2 21.31 X 1/4 1.370.81 Sphere or cylinder?

13. On the small moon of Deimos …. AA’

14. 6. Given that derive an expression for the radius, where I = kF E. Compute the depth to the top of the casing for the anomaly shown below, and then estimate the radius of the casing assuming k = 0.1 and F E =55000nT. Z max (62.2nT from graph below) is the maximum vertical component of the anomalous field produced by the vertical casing.

15. The map view clearly indicates that consideration of two possible origins may be appropriate - sphere or vertical cylinder.

16. In general one will not make such extensive comparisons. You may use only one of the diagnostic positions, for example, the half-max (X 1/2 ) distance for an anomaly to quickly estimate depth if the object were a sphere or buried vertical cylinder…. Burger limits his discussion to half-maximum relationships. Breiner, 1973 X 1/2 = Z/2 X 1/2 = 0.77Z X 1/2 = Z X 1/2 = Z/2

17. Remember how the proton precession magnetometer works. Protons precess about the earth’s total field with a frequency directly proportional to the earth’s field strength The proton precession magnetometer measures the scalar magnitude of the earth’s main field.

18. In this diagram F ET is is the vector sum of the earth’s main field and the anomalous field associated with a buried dipole field. The proton precession magnetometer measures the magnitude of F ET.

19. Recall that the proton precession magnetometer makes measurements of the total field, not the vector components of the field. Recall also that the total field can be derived from other magnetic elements. The formula below represents the anomalous total field in terms of the horizontal and vertical components of the anomalous field.

20. In most applications the anomalous field F A is much smaller than the main field F E.

21. In this case, the magnetic anomaly is approximated as the difference between the measured field (F ET ) at some point and the predicted value of the earth’s main field (F E ) at that point. This anomaly is often referred to as T. 53

22. When F A (the anomalous field) is small, we consider the difference T = F ET - F E to be equivalent to the projection of vector F A onto the direction of the main field – exactly what we did in the derivation of the dipole field. FAFA

23. In the case where F A is large the projection F AT is significantly different from T.

24. Let’s zoom in for a closer look at the tip of F E.

25. Magnetic Elements for your location F E is known

26. Horizontal line parallel to earth’s surface i is the inclination of the earth’s main magnetic field.  is the angle of F A relative to the earth’s main field F E. T F ET

27. F AT is the projection of F A onto the direction of the main field F E, and is considered equal to T, the scalar difference between F E and F ET.

28. The horizontal and vertical projections of F A

29. The horizontal and vertical projections of F A appear in the expansion of F AT = F A cos(  -i).

30. In summary - F AT is an approximation of T, the scalar difference obtained from measurements of the total field (F ET ) made by the proton precession magnetometer. For the purposes of modeling we work backwards. Given a certain object, we compute the horizontal (H A ) and vertical (Z A ) components of the anomaly and combine them to obtain F AT – an approximation of the anomaly we obtain from the proton precession magnetometer measurements – the total field anomaly.

31. Could a total field magnetic survey detect the burial chamber (spherical void) illustrated in the here in a region where F E =55,000nT and i=70 o ? Consider Table 7.3 & Berger’s Table 7-3.xls Finish reading chapter 7 for Tuesday and consider questions 7-5 through 7-9 for in-class discussions and final exam review.

32. The gradient is just the rate of change in some direction - i.e. it’s just a derivative. How would you evaluate the vertical gradient of the vertical component of the earth’s magnetic field?

33. The vertical gradient is just the variation of Z E with change in radius or distance from the center of the dipole.

34. Vertical Gradient

35. Total FieldVertical Gradient http://rubble.phys.ualberta.ca/~doug/G221/MagLecs/magrem.html

36. Representing the earth’s horizontal field in dipole form as The vertical gradient is just the variation with change of radius or Can you evaluate the vertical gradient of the horizontal component of the earth’s magnetic field?

37. You are asked to run a magnetic survey to detect a buried drum. What spacing do you use between observation points?

38. How often would you have to sample to detect this drum?

39. …. how about this one? The anomaly of the drum drops to ½ at a distance = ½ the depth.

40. Remember, the field of a buried drum can be approximated by the field of a dipole or buried sphere. X 1/2 for the sphere (the dipole) equals one-half the depth z to the center of the dipole. The half-width of the anomaly over any given drum will be approximately equal to its depth Or X 1/2 =Z/2

42. The sample rate you use will depend on the minimum depth of the objects you wish to find. X 1/2 = Z/2 X 1/2 = 0.77Z X 1/2 = Z X 1/2 = Z/2

43. Object-centered anomaly

44. Sign Conventions Vectors that point down are positive. Vectors that point south are negative.

45. The analytic “object- centered” signal Original data

47. A 50 foot long drum? Drums in the bedrock?

48. Thursday Dec. 6th Reminder – the final for this class is scheduled for 3 to 5pm on Friday, December 14th