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

2. Model of the theoretical gravity Normal gravity Elevation effect The effect of material beneath the station - the plate effect Topographic or terrain effect Tide and Instrument drift effects

3. More on the plate and terrain corrections - volume element. This remains true in computing the acceleration of a ring. The approach to the terrain correction rests on the analytical expression derived for the acceleration due to gravity of the ring - in particular a given sector of the ring. We start by deriving the acceleration over a disk with 0 inner radius and outer radius R 0. Our starting point should be familiar by now - In dealing with the derivation of the Bouguer plate effect you may have realized that the trick to integration was in how one defined the

4. and assuming that we have constant density throughout the disk

6. Go to 15

7. and again we take the vertical component

9. Notice that we have three differential components, dr, dz, and d , so three separate integrations are implied First consider Some missing steps in the results of integration …….

10. Now take a few minutes and evaluate - Which yields -

11. Substitution yields Use the power rule to obtain -

13. Substitute into to obtain

15. Consider the following- What happens when R 0 goes to  ? g=2  G  (  h) where  h is just the thickness of the plate and could be  z or just z using the notation of the text. This is just the Bouguer plate correction.

16. This is just the plate correction. At infinity the effect of ring and plate are the same.  density contrast t drift thickness    

17. The foregoing approach represents another way to derive the plate correction - and also to determine the effect of a ring.  

18. In the above equation where R o is not , but the outer radius of the cylinder. Rewrite the equation substituting R i for 0, where R i is the inner radius. Note also, that h 1 = 0, and h 2 = z, i.e., the point of observation is right on the top of the cylinder. You could also show that subtracting the acceleration due to the inner smaller cylinder (radius r i ) from the outer cylinder (radius r o ) yields the acceleration of a ring with width r 0 -r i (or R 0 -R i ).

19. Hint - and eventually -

20. Remember we want to approximate topographic features by ring sectors because it’s easy to compute the effect of a ring sector on the observed gravity.

21. In practice, the topography surrounding a particular observation point is divided into several rings (usually A through M) and each ring into several sectors. The F-ring extends from 1280 to 2936 feet and is divided into 8 sectors. The average elevation in each sector is estimated and it’s contribution to the acceleration at the observation point is computed. Let’s spend a few moments working through a simple example to illustrate how the terrain correction is applied.

22. In areas where the terrain is too complex to estimate the average elevation visually, one can compile averages from the elevations observed at several points within a sector. As you might expect - this was a laborious process.

23. Hammer Table

24. What’s the station elevation? What’s the average elevation in Sector 1? What’s the relative difference between the station elevation and the average elevation of sector 1? 2840 feet 2640 feet 200 feet 2600

25. As the legend in the Hammer table notes, the value for T is in hundredths of a milligal and has been calculated assuming a replacement density of 2 gm/cm 3. Thus the contribution to the topographic effect from the elevation differences in this sector is 0.03 milligals. Note that the elevation difference is reported in a range and that the value is not exact for that specific difference - in this case 200 feet. The value could be computed more precisely using the ring formula we developed earlier in class.

26. 20026403 (0.03mG)0.0279 mG For next class - determine the average elevation, relative elevation differenceand T for all 8 sectors in the ring. Add these contributions to determine the total contribution of the F-ring to the terrain correction at this location. Also determine the F-ring contribution if the replacement density of 2.67 gm/cm 3 is used instead of 2 gm/cm 3. Station elevation = 2840 feet This just requires multiplication of the results obtained assuming 2 gm/cm 3 by the ratio 2.67/2 or 1.34. How?

27. Now that we’ve described all the corrections and gained some experience and familiarity with their computation, let’s consider the concept of the gravity anomaly. What is a gravity anomaly? In general an anomaly is considered to be the difference between what you actually have and what you thought you’d get. In gravity applications you make an observation of the acceleration due to gravity (g obs ) at some point and you also calculate or make a prediction about what the gravity should be at that point (g t ). The prediction assumes you have a homogeneous earth - homogeneous in the sense that the earth can consist of concentric shells of differing density, but that within each shell there are no density contrasts. Similar assumptions are made in the computation of the plate and topographic effects. g t then, in most cases, is an imperfect estimate of acceleration. Some anomaly exists. g anom = g obs - g t

28. This is a simple definition, but there are several different types of anomalies, which depend on the degree to which the theoretical gravity has been estimated. For example, in a relatively flat area close to sea-level we might only include the elevation effect in the computation of g t. This would also be standard practice in ocean surveys. In general tide and drift effects are always included. In this case, the anomaly (g anom ) is referred to as the free-air anomaly (FAA).

29. When only the elevation and plate effects are included in the computation of theoretical gravity, the anomaly is referred to as the simple Bouguer anomaly or just the Bouguer anomaly. The combined corrections are often referred to as the elevation correction.

30. When all the terms, including the terrain effect are included in the computation of the gravity anomaly, the resultant anomaly is referred to as the complete Bouguer anomaly or the terrain corrected Bouguer anomaly (  g TBA ).

31. In this form - The different terms in the theoretical gravity are referred to as corrections. Thus -  g FA is referred to as the free-air correction  g B is referred to as the Bouguer plate correction  g T is referred to as the terrain correction, and  g Tide and Drift is referred to as the tide and drift correction

32. There is one additional anomaly we need to discuss. This anomaly is known as the residual anomaly. It could be the residual Bouguer anomaly, or the residual terrain corrected Bouguer anomaly, etc. The issue here is the concept of the residual. Recall from your reading of Stewart’s paper, that he is dealing with the residual Bouguer anomaly. What is it? Most data contain long wavelength and short wavelength patterns or features such as those shown in the idealized data set shown below.

33. Long wavelength features are often referred to as the regional field. The regional variations are highlighted here in green. The residual is the difference between the anomaly (whichever it is) and the regional field.

34. Just as a footnote, we shouldn’t loose sight of the fact that in all types of data there is a certain amount of noise. That noise could be in the form of measurement error and vary from meter to meter or operator to operator. It could also result from errors in the terrain corrections (operator variability) and the accuracy of the tide and drift corrections. If we could separate out the noise (below), we might see the much cleaner residual next to it. Noisy Signal Signal after noise attenuation filtering

35. Stewart makes his estimates of valley depth from the residuals. You shouldn’t be concerned too much if you don’t understand the details of the method he used to separate out the residual, however, you should appreciate in a general way, what has been achieved. There are larger scale structural features that lie beneath the drift valleys and variations of density within these deeper intervals superimpose long wavelength trends on the gravity variations across the area. These trends are not associated with the drift layer. The potential influence of these deeper layers is hinted at in one of Stewart’s cross sections.

36. Bouguer anomaly - Regional anomaly = Residual anomaly

37. If one were to attempt to model the Bouguer anomaly without first separating out the residual, the interpreter would obtain results suggesting the existence of an extremely deep glacial valley that dropped off to great depths to the west. However, this drop in the Bouguer anomaly is associated with the deeper distribution of density contrasts. Let’s spend a few minutes and discuss one method for determining the residual gravity anomaly. The method we will discuss is referred to as a graphical separation method.

38. Examine the map at right. Note the regional and residual (or local) variations in the gravity field through the area. The graphical separation method involves drawing lines through the data that follow the regional trend. The green lines at right extend through the residual feature and reveal what would be the gradual drop in the anomaly across the area if the local feature were not present.

39. The residual anomaly is identified by marking the intersections of the extended regional field with the actual anomaly and labeling them with the value of the actual anomaly relative to the extended regional field. -0.5 After labeling all intersections with the relative (or residual ) values, you can contour these values to obtain a map of the residual feature.

40. Geology 252 Environmental and Exploration Geophysics I In-Class Exercise - Determining residual acceleration  Use the graphical construction approach and estimate the residual anomaly superimposed on the regional gravity gradient.  What is the maximum value of the residual anomaly?  What is the minimum value of the residual anomaly?  Is the anomaly positive or negative? Bring your results to lab

41. Your result will reveal the gravitational effects of an isolated shallow body. It will be easier for you to evaluate the significance of this feature when it is isolated from the regional variations on which it is superimposed. One thing that you should realize and that we will comment on more fully in lab and in later lecture discussions, is the concept of the non-uniqueness of potential field (gravity and magnetic) solutions. The idea is expressed nicely in the following figure taken from Nettleton, 1971

42. Nettleton, 1971 Note that a particular anomaly, such as that shown below, could be attributed to a variety of different density distributions. Note also, however, that there is a certain maximum depth beneath which this anomaly cannot have its origins. gravity anomaly

43. anomaly If there are no subsurface density contrasts - i.e. no geology, then the theoretical gravity equals the observed gravity and there is 0 anomaly. Now let’s consider the significance of the corrected accelerations from a graphical point of view.

44. If there are density contrasts, i.e. if there are materials with densities different from the replacement density, then there will be an anomaly. That anomaly is the geology or site characteristics we are trying to detect.

46. Questions about problem 1? 1. If a gravity determination is made at an elevation of 152.7m, what is the value of the free air correction (assuming that sea-level is the datum)? Also, what is the Bouguer correction assuming a 2.5 gm/cm 3 reduction density? Note that the Bouguer plate and free-air corrections are often combined into the elevation correction - In metric units (mks) Problems taken from Berger (1992)

47. Problem 2: A gravity station at 0m is located in the center of an erosional basin. The floor of the basin has virtually no relief. An escarpment of a plateau is located at a distance of 450 m from the gravity station. The surface of the plateau has a relatively constant elevation of 400m. Will terrain corrections be necessary? Assume that the material above sea level that forms this vast plateau has a density of 2.5gm/cm 3. Note that this problem is nicely set up to employ direct computation of the terrain effect using a ring of inner radius 450m and outer radius of .

48. Problem 3: Prepare a drift curve for the following data and make drift corrections. Convert your corrected data to milligals. The data were collected by a gravimeter with dial constant equal to 0.0869mGals/scale division. In the 110 minute time elapsed between base station measurements g obs has decreased 1.53 scale divisions or 0.133 milliGals. Thus there is a - 0.133mG drift over 110 minutes or a -0.0012 milliGal/minute drift.

49. Let’s take the value at station 3 (GN3). Its value in milliGals relative to the base station is 0.54 milliGals. However, we know that the gravity at the base and therefore throughout the area has been falling at a rate of -0.0012mG/minute. Thus we suspect that the actual difference between the base station and station 3 must be larger. Calculating the drift over a 77 minute period, we find that the gravity at station 3 must, in fact be 77min x 0.0012mG/min or 0.092mG greater than reported.

50. Base 1234 0 -0.133 0.54mG 0.092mG Actual difference relative to the base station is +0.63mG. 0.5

51. Note that we can develop a simple equation t convert relative difference to absolute differences. However, we know that this drop in g through time will increase differences that were initially positive relative to base and decrease values initially negative in relation to the base. Thus our equation should look like Of course it works - look at the final base station measurement

52. I suggest that you let the first measurement at the base station - g base = 0. Also note that there is nothing absolute about 0 scale divisions or, for that matter, the 0 milliGals reference point. There’s no point working with the scale divisions when it is accelerations we are after.

54. Next time - Thursday (Nov. 11 th ) - Hand in problems 1-3. In the next lecture we will cover basic ideas related to the use of simple geometrical objects in gravity interpretation. The extra credit gravity exercise due on Thursday (Nov. 11 th ) Gravity lab is due on Nov. 16 th Gravity paper summaries are due on Nov. 18 th