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. Phobos - Escape velocity 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 You would need a sling shot

3. 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 -

4. Tom Wilson, Department of Geology and Geography 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 a prediction of what the acceleration should be at a particular point on the surface of a homogeneous earth. Terms

5. 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. Anomaly = Lateral density contrast

6. Tom Wilson, Department of Geology and Geography That predicted or estimated value of g is often referred to as the theoretical gravity - g t If the observed values of g behave according to this ideal model then there is no geology! - i.e. there is no lateral heterogeneity. The geology would be fairly uninteresting - a layer cake... We’ll come back to this idea later, but for now let’s develop a little better understanding of the individual terms in this expression. The Theoretical Gravity

7. Tom Wilson, Department of Geology and Geography From our initial discussions you know there are several reasons why g may differ from one point to another on the earth’s surface. 1) The earth is an oblate spheroid, and if we were to walk from the equator to the poles we would go down hill over 21 kilometers. We would be 21.4 kilometers closer to the center of the earth at the poles. The variation in earth radius is primarily a function of latitude. 2) In addition to that we have another latitude dependant effect - centrifugal acceleration. Consider these terms individually- 1 st (g n (  )) – the normal gravity on the ellipsoid

8. Tom Wilson, Department of Geology and Geography Let’s consider the effects of centrifugal acceleration. The velocity of a point on the earth’s equator as it rotates about the earth’s axis is ~ 1522f/s.

9. 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 It’s about 6 times the acceleration of gravity on Phobos. with

10. 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(  )

11. Tom Wilson, Department of Geology and Geography Distance R  decreases with increased latitude

12. 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 Burger et al. 2006

13. 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 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.

14. 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, 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.

15. 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.

16. 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

17. 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

18. 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.

19. 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.

20. Tom Wilson, Department of Geology and Geography Next we estimate the term  g BP - the Bouguer plate term. This term 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

21. 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. The Plate

22. 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

23. 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

24. Tom Wilson, Department of Geology and Geography - obviously we’re not through yet. We now have to carve out those valleys and put the hills back. Then, we compute their influence on g t …. to compensate for the effect of topography on the plate. Valleys and Hills

25. Tom Wilson, Department of Geology and Geography What is the effect of the topography on the observed gravity?

26. On your list … Tom Wilson, Department of Geology and Geography Hand the in-class problems in before leaving today Get started reading and summarizing those gravity papers Read chapter 6 pages 349 to 378; review the remainder of the chapter. Look over problems 6.1 - 6.3 (problems 1-3 in today’s handout). Bring questions to class on Tuesday. Due date November 15 th.

27. List continued … Tom Wilson, Department of Geology and Geography Don’t forget to read Stewart’s paper. The gravity lab is based on his work. Paper is linked on class web page. Next week continue working through the Gravity computer lab assignment handed out today. We’ll continue lab discussions on Tuesday, November 8 th. Gravity lab will be due on November 17 th – just before Thanksgiving break.