7.3 Gravity Anomalies 137 various components to observed gravity with the name of the corrections shown in parentheses:
observed gravity = attraction of the reference ellipsoid
+ effect of elevation above sea level (free-air) + effect of "normal" mass above sea level
(Bouguer and terrain)
+ time-dependent variations (tidal) + effect of moving platform (Eb'tvos)
+ effect of masses that support topographic loads (isostatic)
+ effect of crust and upper mantle density variations ("geology").
(7.12) Our goal is to isolate the last quantity in this summation, the effect of crustal and upper mantle density variations, from all other terms.
Unfortunately, this last quantity is a relatively minor part of observed gravity. The acceleration of gravity at the surface of the earth due to the whole earth is approximately 9.8 m-sec"2 (980 Gal), whereas anomalies caused by crustal density variations are typically less than 10~3 m-sec~2
(100 mGal), less than 0.01 percent of observed gravity. Portable gravity meters are quite capable of measuring gravity to within 10~7 m-sec~2
(0.01 mGal), or about one part in 108, but the various corrections to observed gravity involve assumptions that limit our ability to resolve the geologic component of observed gravity. Depending on a variety of factors, particularly the severity of the surrounding terrain, the actual resolution of the geologic component in field situations may range from 0.1 to 5 mGal.
We will use the simple crustal model shown in Figure 7.3 to help illustrate the various contributions to observed gravity. This cross section includes various examples of lateral variations in density: a topographic edifice, a low-density root that supports the topography in accordance with the principles of isostasy, and a dense body in the upper crust that extends both above and below sea level. Gravity is observed at the topographic surface along a west-east profile, and our goal is to isolate the anomaly caused by just the high-density body in the upper crust.
Exercise 7.3 How can we tell from Figure 7.3 that the profile is directed
9H0JHX) H Observed Gravity
i
= 297()kg/m3
-60 -40 -20 0 20 40 60
Distance, km
Fig. 7.3. Crustal cross section to describe various corrections to observed grav- ity. The crust and mantle are assumed to have densities of 2670 and 3070 kg-m~3, respectively. The mountain range is isostatically compensated by a crustal root. A mass of rectangular cross section and density 2970 kg-m~3 represents a density variation due to upper-crustal geology. Vertical exagger- ation 2.
The first correction described by equation 7.12 is easily accomplished with the results of the previous section. Equation 7.11 provides theoret- ical gravity, the normal gravitational attraction of a hypothetical earth containing no lateral density inhomogeneities. When this equation is evaluated and subtracted from gravity measurements, the remainder re- flects departures of the earth's density from the homogeneous ellipsoid, in particular lateral density variations in the crust and mantle. The
7.3 Gravity Anomalies Observed Gravity -
Theoretical Gravity
139
-60 -40 -20 0 20
Distance, km
Fig. 7.4. Crustal cross section of Figure 7.3 after subtraction of theoretical gravity. The large negative anomaly is caused primarily by increasing distance between the gravity meter and the reference ellipsoid as the profile rises over the topographic edifice.
remainder also includes the effects of altitude, tides, and various other factors, and these will be discussed subsequently.
Figure 7.4 shows how the crustal cross section of Figure 7.3 is effec- tively changed by subtraction of theoretical gravity. The resulting grav- ity profile is dominated by a large negative anomaly caused primarily by the increasing altitude of the gravity meter as the profile goes over the topographic edifice. This contribution obviously is not related directly to crustal or mantle sources; it merely reflects changes in distance between the gravity meter and the center of the earth.
7.3.1 Free-Air Correction
Shipboard gravity measurements can be compared directly with the ref- erence field go because the geoid corresponds to sea level. Gravity mea- surements over land, however, must be adjusted for elevation above or below sea level. Let g(r) represent the attraction of gravity on the geoid.
The value of gravity a small distance h above the geoid is given by a Taylor's series expansion,
g(r + h) = g{r) + h —g(r) + • • • .
Dropping high-order terms and rearranging the remaining terms gives g(r) =g(r + h)-h 7 ^ # 0 ) •
If we assume that the earth is uniform and spherical, then g(r) =
—'jM/r2, and the previous equation becomes
The last term of this equation accounts for the difference in elevation between g(r) and g(r -\- h). It is known as the free-air correction #fa because it is the only elevation adjustment required if no masses were to exist between the observation point and sea level. Using values of g and r at sea level provides
0fa = -0.3086 x 10~5 h, (7.13) where h is height above sea level. Equation 7.13 is the same in both SI units (#fa in m-sec~2, h in m) and cgs units (#fa in Gal, h in cm) because gfa/h has units of sec"2. Application of the free-air correction provides the free-air anomaly given by
A#fa = gobs ~ #fa - £o , (7.14) where gobs is observed gravity. It should be clear that shipboard mea- surements minus go are at once free-air anomalies.
Figure 7.5 shows the effect of the free-air correction on the hypo- thetical cross section of Figure 7.3. The large negative anomaly of the previous figure, which was caused by increasing elevation of the gravity meter over the topographic high, has been eliminated by the free-air correction. Over elevated areas of land, the free-air anomaly tends to rise to large values, which causes an often undesirable correlation be- tween topography and gravity. This is apparent in Figure 7.5, where
7.3 Gravity Anomalies
Free-Air Anomaly
141
-60 -20 0 20
Distance, km
60
Fig. 7.5. Crustal cross section of Figure 7.3 after the free-air correction. Note that the observation points are not "moved to sea level" and that the free-air anomaly is strongly influenced by terrain.
although the free-air correction has accounted for the variation in el- evation of the measurements, it has not accounted for the additional mass represented by the topographic edifice. Notice too that the crustal root in Figure 7.5, which isostatically supports the topography, also pro- duces a long-wavelength, relatively low-amplitude, negative component in the free-air anomaly. Nevertheless, free-air anomalies are often used in geodesy for studies of the spheroid and geoid because they are very nearly equivalent to what would be observed if all the topographic masses were condensed onto the geoid.
Exercise 7.4 Sketch in profile form the free-air anomaly that would be