The "Normal" Earth

Exercise 6.3 Give an example of a spherical harmonic function of degree 4

7.2 The "Normal" Earth

Observed field

Assumed regional field

Fig. 7.1. The definition of regional field depends on the scale of the problem.

Application of these various operations, if they are successful, will leave us with a gravity anomaly that reflects density variations in the crust and upper mantle. But even this product may not be quite what is needed. Our eventual objective may be the analysis of just one geologic element of the crust, such as a sedimentary basin or some plutonic com- plex, and the anomaly due to this one element ideally should be isolated from those of the surrounding geologic environment. This final aspect of regional-residual separation is often a subjective process and ultimately a matter of scale. A geophysicist attempting to estimate chromite po- tential within a buried ophiolite will probably consider the anomalies due to surrounding sources to be a nuisance, whereas another geophysi- cist studying the relationships between accreted terrains might regard as noise the anomaly due to the chromite (Figure 7.1).

7.2 The "Normal" Earth

Because the force of gravity varies from place to place about the earth, equipotential surfaces surrounding the earth are smooth but irregular.

An equipotential surface of particular interest is the geoid, the equipoten- tial surface described by sea level without the effects of ocean currents, weather, and tides. The geoid at any point on land can be thought of as the level of water in an imaginary canal connected at each end with an ocean. The shape of the geoid is influenced by underlying masses; it

bulges above mass excesses (e.g., mountain ranges or buried high-density bodies) and is depressed over mass deficiencies (e.g., valleys or buried low-density bodies). Because the geoid is an equipotential surface, the force of gravity at any point on the geoidal surface must be perpen- dicular to the surface, thereby defining "vertical" and "level" at each point.

Because of the complexity of internal density variations, it is cus- tomary to reference the geoid to a simpler, smoother surface. By inter- national agreement, that equipotential surface is the spheroidal surface that would bound a rotating, uniformly dense earth. Differences in height between this spheroid and the geoid are generally less than 50 m and re- flect lateral variations from the uniform-density model. The shape of the reference spheroid was first investigated by measuring the arc lengths of degrees at various latitudes. It was recognized by the late 1600s that the spheroid is oblate (see Introduction). In fact, because of the competing forces of gravity and rotation, the spheroid very nearly has the shape of an ellipse of revolution and, consequently, is called the reference ellip- soid. It should be intuitive in any case that the spheroid is symmetric through its center and symmetric about the axis of rotation. Its shape is described by just two parameters, the equatorial radius a and polar radius c (Figure 7.2), and often is expressed in terms of the flattening parameter

The earth is nearly spherical, of course, with flattening of only 1/298.257, and this fact will permit several simplifying approximations in the fol- lowing derivations.

The force of gravity on the earth is due both to the mass of the earth and to the centrifugal force caused by the earth's rotation. The total potential of the spheroid, therefore, is the sum of its self-gravitational potential Ug and its rotational potential Ur,

U = Ug + Ur, (7.1)

where

UY = \u2r2 cos2 A,

uo is angular velocity, and A is latitude (Heiskanen and Moritz [123]).

Exercise 7.1 Show that UT is the potential of a centrifugal force fr, in the sense that it satisfies f^r = VC/^r. Is U^T harmonic?

7.2 The "Normal" Earth

CO

131

Fig. 7.2. Parameters involved in describing reference ellipsoid.

The gravitational potential Ug is harmonic outside the spheroid, and according to Section 2.1.1, is uniquely determined everywhere outside by its values on the surface. As we shall see shortly, Ug on the surface is determined completely by /, a, and the total mass of the earth. Hence, just these three parameters plus uo are sufficient to find the total potential

U of the spheroid anywhere on or above its surface.

The self-gravitational potential is given by equation 6.31,

n=0

n = 0

, (7-2)

ra=O

where M is total mass, a is equatorial radius, 4> is longitude, and 6 is colatitude. Equation 6.31 was derived in Chapter 6 from Laplace's equation, V2V = 0, in spherical coordinates with no particular physical meaning attached to V. In equation 7.2, however, the various terms in the expansion describe the gravitational potential in terms of an infinite set of idealized masses (monopole, dipole, and so forth) centered at the origin, the coefficients a™ and /?™ describing the relative importance of each mass.

Symmetry of the spheroid greatly simplifies this equation. First, Ug

has no dependence on 0, so all terms with m ^ 0 are zero. Therefore, with the help of Table 6.2, gravitational potential reduces to

Lo ^{+ a}o «c o s 0 + ao (*\ i o g 2 0 + 1 } +. . 1 ^{( 7 3 )}

The first term of this equation is the monopole term, which must equal 7M/V. Hence, a$ = 1. The second term, the dipole term, must be zero because the origin is at the center of mass. Hence, a\ — 0, and all other coefficients of odd degree must be zero for the same reason. Con- sequently, the third term is the lowest term in the series that describes the departure of the spheroid from a sphere. The coefficient a® is gener- ally expressed in terms of the ellipticity coefficient J2, where a® = —Ji- Its relationship to the flattening / of the spheroid is given approximately by

2f-m

= 1.082626 x 1(T3

(Stacey [270, p. 90]), where m is the ratio of the centrifugal force to the gravitational force at the equator, given by

m =-fM/a2

_oo2a3

~ -yM

= 3.46775 x 10"3 .

Dropping all higher terms in equation 7.3, changing from colatitude to latitude, and substituting U^g into equation 7.1 yields the total gravita- tional potential

Jf M£J (3 sin2 A - 1) + \jr> cos2 A. (7.4)

If the spheroid is approximately spherical, any normal to the spheroid will be very nearly parallel to r. Then total gravity, normal to the

7.2 The "Normal" Earth 133 spheroid and directed inward, is given approximately by

dU go — —-£-

<yM 3 7 M Qor ²J² . ^{2 2 2}

= —^ -: (3 sm A — 1) — UJ r cos A , (7.5) rz 2 r4

where go is used here to denote the total gravity of the spheroid. Note that in previous chapters, the radial component of gravity was defined as gr = |^7, so the force of gravity was negative in the direction of increasing distance from the spheroid. Equation 7.5 differs from this sign convention, and go is always positive.

Equation 7.5 describes the total gravity of the ellipsoid anywhere on or outside the ellipsoid in a reference frame that moves with the spin of the earth. If we can express r in this equation in terms of a and A, we can obtain a simplified view of how total gravity varies on the surface of the ellipsoid. The radius of an ellipsoid is given approximately by the relation

r = a ( l - / s i n2A ) . (7.6) Exercise 7.2 Prove the previous statement. Hint: The defining equation for an ellipse, x2/a2 + y2/b2 — 1, is a good place to start, and the binomial expansion provides a useful approximation.

Because / is small, we can use this expression to expand 1/r2 in a binomial series,

i_ = 1 ( 1 + 2/ sin2 A),

and substitute into the first term of equation 7.5. The last two terms of equation 7.5 are sufficiently small relative to the first term that the approximation r = a will suffice. Making these substitutions leads to

go = _4_(i + 2/sin2 A) - - V J2 (3sin2 A - 1) - u2a(l - sin2 A) cr 2 or

2f-^J2+m\sm2\\ • (7.7) At the equator, equation 7.7 becomes

h - J3 2-

and substituting this expression into equation 7.7 and rearranging terms provides a simple relation describing the total gravitational attraction of the spheroid,

<7o = <7e(l + /'sin2A), (7.8) where

Of

2/-§J

²

Equation 7.8 has the same form as equation 7.6, namely, that of an ellipse. Hence, to first order, the total gravity of the spheroid varies with latitude as the radius of a prolate ellipsoid. At the pole,

g^p = g^e(l + / ' ) , so

J 5

and the parameter / ' in equation 7.8 is the gravitational analog of geo- metrical flattening.

The parameters ge, #p, and / ' have values of 9.780327 m-sec~2, 9.832186 m-sec~², and 0.00530, respectively, and as we should have ex- pected, the total gravity of the reference ellipsoid varies by only a small amount over its surface, about 0.5 percent from equator to pole. As we shall see, however, this small variation is nevertheless significant when compared to gravitational attraction of geologic sources.

Theoretical Gravity

Carrying through the previous derivations to higher order, equation 7.8 can be cast more accurately as

go = g^e(l + a sin² A + /3sin² 2A), (7.9) where, as before, ge is the equatorial attraction of the spheroid, and a and (5 depend only on M, / , a;, and a. Equation 7.9 is a truncated infinite series, but a closed-form expression for go can be derived as well (Heiskanen and Moritz [123, p. 70]),

/ l + /csin2A \

g0 = ge[ , 7 . 1 0

V\/l-e

²

sin

²

Ay

where k and e also depend only on M, / , CJ, and a. This equation is called the Somigliana equation.

7.2 The "Normal" Earth 135 Table 7.1. Parameters of various geodetic reference systems, from

Chovitz [58].

System a, km / J2 7M, m3-sec~2 ge, m-sec^{- 2} 1924-30 6378.388 1/297.0 0.0010920 3.98633xl0¹⁴ 9.780490 1967 6378.160 1/298.247 0.0010827 3.98603xl014 9.780318 1980 6378.137 1/298.257 0.00108263 3.986005xl0¹⁴ 9.780327

Hence, the gravitational attraction of the reference ellipsoid at any point (r, A), whether expressed by equation 7.9 or 7.10, depends on only four observable quantities: 7M, a, J2 (or / ) , and uo. The quantity 7 M is considered one parameter here because the product of 7 M can be de- termined much more precisely than either 7 or M separately. The equa- torial radius a is found from arcs of triangulation, and rotation velocity UJ is found from astronomical measurements. Prior to the first artificial satellites, 7 M and J2 were based on surface gravity measurements. Now 7 M and J2 are found from satellite observations and planetary probes (Chovitz [58]). Note that detailed knowledge of the earth's density is not required in order to specify the ellipsoid.

As knowledge of the defining parameters have evolved over recent years, so too has the reference ellipsoid. The ellipsoid is defined and refined by international agreement through the International Association of Geodesy (IAG) and its umbrella organization, the International Union of Geodesy and Geophysics (IUGG). Three international systems have been sanctioned in this way, and Table 7.1 shows the defining parameters for each of these systems. The first internationally accepted reference ellipsoid was established in 1930, and its associated parameters provided the 1930 International Gravity Formula,

go = 9.78049(1 + 0.0052884sin2 A - 0.0000059 sin2 2A),

where go is in m-sec~2. The advent of satellites provided a breakthrough in the accuracy of various geodetic parameters, and a new ellipsoid was adopted in 1967 called Geodetic Reference System 1967, thereby provid- ing the 1967 International Gravity Formula,

go = 9.78031846(1 + 0.0053024 sin² A - 0.0000058 sin² 2A).

Most recently the IAG has adopted Geodetic Reference System 1980, which eventually led to the current reference field, World Geodetic

System 1984; in closed form it is given by

,o = — 1 + 0-00193185138639^^

VI - 0.00669437999013 sin² A

The quantity go, expressed by equation 7.11 or its predecessors, is com- monly referred to as theoretical gravity or normal gravity.

The Geoid

As discussed previously, the reference ellipsoid is the equipotential sur- face of a uniform earth, whereas the geoid is the actual equipotential sur- face at mean sea level. Differences in height between these two surfaces rarely exceed 100 m and generally fall below 50 m (Lerch et al. [163]).

The shape of the geoid is dominated by broad undulations, with lateral dimension of continental scale but with no obvious correlation with the continents; they apparently are caused by widespread mantle convection (Hager [108]). Compared with these broad undulations, the response of the geoid to topography and density variations within the lithosphere are second-order effects, both low in amplitude and short in wavelength (Marsh et al. [175]; Milbert and Dewhurst [184]).

Gravity anomalies, to be discussed in the next sections, are referenced to the reference ellipsoid but involve various corrections relative to sea level (the geoid). This inconsistency is ignored in most crustal studies, and in the following discussion, we too will assume that go represents theoretical gravity on the geoid. While this implicit assumption is ac- ceptable for most geologic studies, the discrepancy between the reference ellipsoid and the geoid should be accounted for if the size of the study is on the order of the broad-scale undulations of the geoid.

7.3 Gravity Anomalies

The isolation of anomalies caused by local density variations from all other fields involves a series of corrections to observed gravity. They can be confusing to students because of the way they sometimes are described. For example, the free-air correction, to be discussed subse- quently, is sometimes inaccurately described as "moving the observation point downward to sea level." It would be incorrect, however, to consider the observation point at sea level in subsequent calculations or graphical displays.

A better way to describe the series of corrections is to consider them each as contributors to observed gravity. The following sum shows the

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