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

2. Reminders Tom Wilson, Department of Geology and Geography Resistivity lab due today (Thursday October 15 th ) Also, no class, October 20 th and 22 nd. Gravity papers are out. Since next week is off, get a paper and write that summary. No due date yet, but good use of time. Will take questions today on intro problems 1-3. Turn them in when you can. Just put in my mailbox. Writing section outline for Essay II is due in my mailbox October 23 rd.

3. Questions Tom Wilson, Department of Geology and Geography Let’s make these due in my mailbox sometime in the next week by end-of-day October 23 rd.

4. For today Tom Wilson, Department of Geology and Geography Additional geological examples Discrete integration concepts g produced by a buried equidimensionally shaped object Factors influencing g Free air influence Influence of topographic features Latitude effect The influence of solar and lunar tides Instrument drift Calculating the theoretical gravity The gravity anomaly

5. Another good use of your time – read over this paper Tom Wilson, Department of Geology and Geography

6. Form Stewart Bedrock models derived from gravity data Residual gravity data The gravity anomalies associated with these glacial valleys have a range of about 4 milliGals. Why residual? The residual eliminates the influence of the deeper strata which dip uniformly across the area. Their configuration is not relevant to the problem at hand. The residual can eliminate geology we aren’t interested in

7. Teays valley pre-glacial river channel Tom Wilson, Department of Geology and Geography 2.5 milliGal gravity low

8. Tom Wilson, Department of Geology and Geography Karst 0.5 milliGal gravity low over karst collapse feature compared to terrain conductivity response

9. Tom Wilson, Department of Geology and Geography -76 mGals -32 mGals The gravity anomalies in West Virginia are related to an old plate suture and mountain root.

10. The acceleration term in Newton’s universal law of gravitation Tom Wilson, Department of Geology and Geography tells us we need to consider mass (m) and its distance(s) (r i ) from some observation point. In practice we usually compute the acceleration of some arbitrarily shaped mass by breaking it up into small parts and summing their individual contributions to g.

11. Tom Wilson, Department of Geology and Geography Integral form of Newton’s law of gravitation Line, surface or volume Depending on symmetry dz dy dx dV

12. Tom Wilson, Department of Geology and Geography Just as a footnote, Newton had to develop the mathematical methods of calculus to show that spherically symmetrical objects gravitate as though all their mass is concentrated at their center.

13. Tom Wilson, Department of Geology and Geography We will take advantage of this when calculating the gravitational attraction of a buried equidimensional (roughly spherically symmetrical) region of density contrast? We’ll work this out in some detail later

14. Also note that the gravimeter measures the vertical component of g Tom Wilson, Department of Geology and Geography

15. If the Earth maintained spherical symmetry at all scales, our work would be done Tom Wilson, Department of Geology and Geography Lateral density contrasts would not exist and there would be no gravity anomalies.

16. Tom Wilson, Department of Geology and Geography How thick is the landfill? Gravity methods thrive on heterogeneity. In general the objects we are interested in are not so symmetrical and provide us with considerable lateral density contrast and thus gravity anomalies.

17. How does g vary from point A to E across the landfill? Tom Wilson, Department of Geology and Geography We might expect that the average density of materials in the landfill would be less than that of the surrounding bedrock and thus be an area of lower g, where g would vary in proportion to 

18. Tom Wilson, Department of Geology and Geography These variations in gravitational acceleration are very small. To give you some additional perspective on the magnitude of these changes, consider the changes in g as a function of r (or R E ) as indicated by Newton’s law of gravity - Recognize that the above equation quantifies the variation in g as a function of r for objects that can effectively be considered as points. For now, let’s take a leap of faith and assume that we can represent the Earth as a point and that the above equation accurately describes the variations in g as a function of distance from the center of the earth, R E. How does g vary with distance R from the center of the Earth?

19. Tom Wilson, Department of Geology and Geography Given this relationship - RERE h What is g at a distance R E +h from the center of the earth? sl=sea level

20. Computation of g at an elevation h above sea level Tom Wilson, Department of Geology and Geography Is there another way to compute the change in g?

21. Tom Wilson, Department of Geology and Geography What does the derivative of g with respect to R provide?

22.  g/  h - in Morgantown Tom Wilson, Department of Geology and Geography At Morgantown latitudes, the variation of g with elevation is approximately 0.3086 milligals/m or approximately 0.09406 milligals/foot. As you might expect, knowing and correcting for elevation differences between gravity observation points is critical to the interpretation and modeling of gravity data. The anomalies associated with the karst collapse feature were of the order of 1/2 milligal so an error in elevation of 2 meters would yield a difference in g greater than that associated with the density contrasts around the collapsed area.

23. Tom Wilson, Department of Geology and Geography How do we compensate for the influence of matter between the observation point (A) and sea level? How do we compensate for the irregularities in the earth’s surface - its topography? A hill will take us down the gravity ladder, but as we walk uphill, the mass beneath our feet adds to g.

24. Tom Wilson, Department of Geology and Geography What other effects do we need to consider? Latitude effect 463 meters/sec ~1000 mph Centrifugal acceleration carries you around the Earth with velocity

25. Tidal influence/instrument drift Tom Wilson, Department of Geology and Geography Solar and Lunar tides Instrument drift

26. Making a prediction of g at any location on Earth Tom Wilson, Department of Geology and Geography To conceptualize the dependence of gravitational acceleration on various factors, we usually write g as a sum of different influences or contributions. These are -

27. Terms in the predicted or theoretical g at a given location. Tom Wilson, Department of Geology and Geography Terms include: g n the normal gravity of the gravitational acceleration on the reference ellipsoid  g FA the elevation or free air effect  g B the Bouguer plate effect or the contribution to measured or observed g of the material between sea-level and the elevation of the observation point  g T the effect of terrain on the observed g  g Tide and Drift the effects of tide and drift (often combined) These different terms can be combined into an expression which is equivalent to the prediction of what the acceleration should be at any particular observation point on the surface of a homogeneous earth.

28. You end up with a predicted value of what g should be in the absence of subsurface density contrasts. Tom Wilson, Department of Geology and Geography Thus when all these factors are compensated for, or accounted for, the remaining “anomaly” is associated with lateral density contrasts within area of the survey. The geologist/geophysicist is then left with the task of interpreting/modeling the anomaly in terms of geologically reasonable configurations of subsurface intervals.

29. The theoretical or predicted acceleration as a formula Tom Wilson, Department of Geology and Geography That predicted, estimated or theoretical value of g, g t, (or predicted value, g p, they are the same) is expressed as follows: If the observed values of g behave according to this ideal model (i.e. if g o =g p ) then there is no complexity in the geology! - i.e. no lateral density contrasts. The geology would be fairly uninteresting - a layer cake... Let’s look at the individual terms in this expression.

30. Tom Wilson, Department of Geology and Geography What is the centrifugal acceleration at the equator? Although that acceleration is small, if you were subjected only to that acceleration, you would fall 0.4 meters in 5 seconds 1.65 meters in 10 seconds The centrifugal acceleration alone is about 6 times the acceleration of gravity on Phobos. with

31. Tom Wilson, Department of Geology and Geography Note that as latitude changes, R in the expression does not refer to the earth’s radius, but to the distance from a point on the earth’s surface to the earth’s axis of rotation. This distance decreases with increasing latitude and becomes 0 at the poles.  R(  )

32. Tom Wilson, Department of Geology and Geography Distance R  decreases with increased latitude At the poles, you turn on the spot once every 24 hours.

33. Tom Wilson, Department of Geology and Geography The combined effects of the earth’s shape and centrifugal acceleration are represented as a function of latitude (  ). The formula below was adopted as a standard by the International Association of Geodesy in 1967. The formula is referred to as the Geodetic Reference System formula of 1967 or GRS67 g n (  ) incorporates latitude effects See page 357, eqn. 6.12 - Burger et al. 2006

34. Familiarize yourself with the units discussed earlier Tom Wilson, Department of Geology and Geography Remember the gravity unit (otherwise known as gu)? Recall that the milliGals represent 10 -5 m/sec 2 The milliGal is referenced to the Gal. In recent years, the gravity unit (gu) has become popular, largely because instruments have become more sensitive and it’s reference is to meters/sec 2 i.e. 10 -6 m/sec 2 or 1 micrometer/sec 2.

35. Tom Wilson, Department of Geology and Geography The gradient of this effect is This is a useful expression, since you need only go through the calculation of GRS67 once in a particular survey area. All other estimates of g n can be made by adjusting the value according to the above formula. The accuracy of your survey can be affected by an imprecise knowledge of one’s actual latitude. The above formula reveals that an error of 1 mile in latitude translates into an error of 1.31 milliGals (13.1 gu) at a latitude of 45 o (in Morgantown (about 40 o N, this gradient is 12.84 gu/mile). The accuracy you need in your position latitude depends in a practical sense on the change in acceleration you are trying to detect.

36. Tom Wilson, Department of Geology and Geography The difference in g from equator to pole is approximately 5186 milligals. The variation in the middle latitudes is approximately 1.31 milligals per mile (i.e. sin (2  ) = 1). Again, this represents the combined effects of centrifugal acceleration and polar flattening.

37. Tom Wilson, Department of Geology and Geography The next term in our expression of the theoretical gravity is  g FA - the free air term. In our earlier discussion we showed that dg/dR could be approximated as -2g/R. Using an average radius for the earth this turned out to be about 0.3081 milligals/m (about 3 gu).  g FA - Free air term

38. Ignore the last two terms and multiply both sides by  R (or h) to get  g Tom Wilson, Department of Geology and Geography When the variations of g with latitude are considered in this estimate one finds that Where z is the elevation above sea-level. The influence of variation in z is actually quite small and generally ignored (see next slide). i.e. for most practical applications  g=-0.3086  R milligals/m Berger et al. Formula 6.14, p 359 For our work in this class we ignore these terms The  R corresponds to z or h as used in earlier discussions

39. Tom Wilson, Department of Geology and Geography As the above plot reveals, the variations in dg/dR, extend from -0.30883 at the equator to -0.30837 at the poles. In the middle latitude areas such as Morgantown, the value -0.3086 is often used. Note that the effect of elevation is ~ +2/100,000th milliGal (or 2/100ths of a microgal) for 1000 meter elevation. Free Air Effect = From Burger et al.

40. Tom Wilson, Department of Geology and Geography The variation of dg/dR with elevation - as you can see in the above graph - is quite small. From Burger et al. So, again, for our work in class, we will ignore the second and third terms and calculate the change in acceleration with change in elevation as

41. Compensating for matter between your observation point and sea level Tom Wilson, Department of Geology and Geography This is a two-step process that involves treating the influence of materials beneath the surface as resulting from a featureless flat blanket of material having a thickness equal to the elevation above sea level, with the influence of topographic features introduced in a second step. Two corrections are required: the plate correction and the topographic correction.

42. Tom Wilson, Department of Geology and Geography  g BP estimates the contribution to the theoretical gravity of the material between the station elevation and sea level. We have estimated how much the acceleration will be reduced by an increase in elevation. We have reduced our estimate accordingly. But now, we need to increase our estimate to incorporate the effect of materials beneath us. First we consider the plate effect from a conceptual point of view and then we will go through the mathematical description of this effect.  g BP – the Bouguer Plate Term

43. Removing the influence of the “plate” Tom Wilson, Department of Geology and Geography GRS67 makes predictions (g n) of g on the reference surface (i.e. sea level). If we want to compare our observations to predictions we have to account for the fact that at our observation point, g will be different from GRS67 not only because we are at some elevation h above the reference surface but also because there is additional mass between the observation point and the reference surface along with the potential for additional lateral density contrasts.

44. Mixing units is possible if you do it right! Tom Wilson, Department of Geology and Geography Thus g plate = 4.192 x 10 -7 cm/s 2 (or gals) for a t = 1 cm and  = 1gm/cm 3. This is also 4.192 x 10 -4 mgals since there are 10 3 milliGals per Gal. Also if we want to allow the user to input thickness (t) in meters, we have to introduce a factor of 100 (i.e. our input of 1 meter has to be multiplied by 100) to convert the result to centimeters. This would change the above to 4.192 x 10 -2 or 0.04192. Where density is in gm/cm 3 and t is in meters also written asSee eqn6.24 in the text.

45. Tom Wilson, Department of Geology and Geography  g B may seem like a pretty unrealistic approximation of the topographic surface. It is! You had to scrape off all mountain tops above the observation elevation and fill in all the valleys when you made the plate correction. Topographic effects See figure 6.3

46. Tom Wilson, Department of Geology and Geography So - now we have to carve out those valleys and put the hills back. We compute their influence on g t …. to compensate for the effect of topography on the plate. Valleys and Hills

47. What is the effect of a topographic feature such as a hill or valley Tom Wilson, Department of Geology and Geography In either case, the influence is to decrease the local acceleration due to gravity

48. Could you jump into orbit on the small moon Phobos? Tom Wilson, Department of Geology and Geography Phobos has a mass of 1.08 x 10 16 kg It has an average radius of about 11.1km V escape =11.4 m/s 100 meters in 8.77s World record 9.6 m/s You would need a sling shot

49. Tom Wilson, Department of Geology and Geography Let’s delay problems (1-3) just have them in my box on the 23rd. Resistivity lab due today Writing section: essay 2 outline due via email ~Oct. 20 th 1 st draft essay due November 3 rd Look over problems 6.1 through 6.3 as handed out in class last time and bring questions to classTuesday, 27th. Read over Stewart’s paper in preparation for the gravity lab effort. No class October 20 th and 22 nd.