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2.1 Gravity and the gravity field of the Earth

Fundamentals

The mass and shape of the Earth

The Geoid

Satellite measurements of gravity

GRACE - Gravity Recovery and Climate Experiment

Fundamentals

The force that attracts all matter to the Earth is the gravity field; an example

of the general phenomenon of attraction between any masses. Newton's Law

describes the force of attraction between two point masses, M1 and M2 separated by

r:

1 2

2

M MF G

r

(2.1.1)

The force per unit mass, F/M2 defines the vector gravity field or the

gravitational acceleration, g when M1 is the Earth (Me) and r is the radius of the

Earth, Re. So

F

M1

= g = GM

e

Re

2r

(2.1.2)

[Note that vectors are denoted either in boldface text or with an overlying arrow, e.g.

r .]

The gravitational constant G has been determined experimentally and is:

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6.67259 x 10-11

m3 kg

-1 s

-2 ( SI units )

or

6.67259 x 10-8

cm3 g

-1 s

-2

with an uncertainty of 100 parts per million (ppm).

The gravitational acceleration is also measured experimentally and at the

equator is about 9.8 meters/sec/sec (m s-2

) or 980 cm s-2

. (In almost all gravity

surveys results are presented in c.g.s units rather than SI units. In c.g.s the unit

acceleration, 1.0 cm s-2

, is called the gal (short for Gallileo). A convenient subunit

for surveys is the milligal, mgal, 10-3

cm s-2

. Another unit that has been used is the

gravity unit, gu, which is defined as 10-6

m s-2

or 0.1 mgal.)

Some properties of the gravitational force field are required for explaining the

variations in g that are observed on and above the Earth's surface and for interpreting

these variations in terms of the desired distributions of densities in the subsurface. A

test mass M gains potential energy U in being moved a distance S in the

gravitational force field F according to:

B B

A AU F dl M g dl (2.1.3)

Figure 2.1.1

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For example, moving a test mass vertically along the radius for uniform

spherical earth the difference in potential energy is:

2 2 2

1 1 12

2 1

1R R Re e

eR R R

GM GMFdr g dr GM dr

M R Rr

the function eGM

UR

is the potential of that point in space. All points in space

that have the same potential define an equipotential surface. By definition the force

field is everywhere perpendicular to the equipotential surface. Potentials cannot be

measured directly, only their force fields. Potential fields are conservative meaning

that the work done in moving from A to B in Figure 2.1.1 is independent of the path

taken. It is obvious from the simple example of the force between two masses that

the force Mg is simply the derivative of the potential in the direction between the

masses. In general, for any conservative field;

g U (2.1.4)

Potentials are additive so that for any point P external to a volume V of

matter:

3

'' '

PV V

r d rGdmU r G

r r r r

(2.1.5)

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Figure 2.1.2

This is the basic formula used to calculate the potential of any subsurface density

distribution. [Note that this integral has the form of a convolution of the density (the

source function) and the mathematical form of the fall off with distance of the field

from a source point (the Green’s function). This concept will be used later to derive

a fast and efficient method to calculate the gravity anomalies from subsurface

bodies.]

Another property of this conservative force field is that the integral of the

normal component of the gravitational acceleration over the surface S of the volume

containing mass M satisfies Gauss's law:

4 4V V

g n dS G dm G dV (2.1.6)

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This relationship is easy to verify for a sphere of uniform density, but

Equation 2.1.6 is more general, and applies to any volume with arbitrary density. We

will use this integral later in the section on interpretation to determine the excess

mass of any finite body in the subsurface.

The divergence theorem states that:

g ×n dsSò = Ñ×g dV

Vò (2.1.7)

and g U so,

4V

U G dV

or

2 4

v VU G dV

In any region where no matter is present, for example in the space above the

earth, this leads to:

2 0U Laplace’s equation (2.1.8)

The mass and shape of the Earth

The mass of the earth can be estimated through Gauss’ law:

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4 4 e

V

g n dS G dV GM (2.1.9)

The radius of the earth Re was determined remarkably accurately by

Eratosthenes in the third century B.C. using the observations shown in the sketch of

Figure 2.1.3. This was arguably the first geophysical measurement.

Figure 2.1.3

It is now known even more accurately through geodetic measurements. Assuming a

spherical earth with a mean value of observed g, the experimental measurements of

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G, and the radius, the mass of the Earth is found from 24 e eg R GM to be

5.976 × 1024

kg.

If there were a perfect sphere of uniform density the gravity field measured on its

surface would be the simple Newtonian field of a point mass at the center of the

earth plus the radial acceleration term caused by the rotation of the earth as shown in

Figure 2.1.4.

Figure 2.1.4

The Earth is not a perfect sphere and it is very close to the shape that a

spinning fluid would have. Because of the rotation the Earth is flattened as shown in

Figure 2.1.5 and to first order the shape is that of an oblate spheroid or ellipsoid.

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Figure 2.1.5

Any section through the axis of rotation is an ellipse in which the minor axis is the

polar diameter and the major axis is the equatorial diameter. The flattening is

described by the flattening parameter:

e p

e

R Rf

R

(2.1.10)

Early measurements of f were made geodetically, that is by measuring the varying

length on the surface of a degree of latitude. More accurate measurements are now

made using satellites. The orbit of an earth satellite over a spherical earth of uniform

radial density distribution is a stationary ellipse. Because of the bulge associated

with the flattening the extra ring of mass exerts a pull or torque on any orbit that is

inclined to the equatorial plane. Accurate measurement of the resulting rotation of

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the plane of the satellite orbit, known as the precession, leads directly to an estimate

of the flattening [See the classic text by Kaula (1960)].

The Geoid

Because the Earth is rotating the observed gravity field has an added

component due to the centrifugal acceleration. At point A on the surface at latitude

and radius r, Figure 2.1.4, the centrifugal acceleration, grot, in the direction normal

to the surface is given by:

2 2 2cos cos cosrotg R R

where is the angular frequency of rotation.

This centrifugal grot is in the opposite direction to the gravitational g so the observed

g is given by:

2 2cosobs zg g r

The value of gz on the oblate spheroid is not a simple calculation and the credit for

the solution, including the rotation term, is due to Clairaut. Clairaut solved the

problem for the gravitational potential on the surface of the oblate spheroid. This

surface is in fact an equipotential surface; the gravitational acceleration vector is

everywhere perpendicular to this surface. The actual surface of this equipotential is

defined as the geoid. The ideal surface derived from the symmetric ellipsoid is

called the reference ellipsoid and it does not coincide exactly with the actual geoid

because the Earth is not homogeneous in density and there are deformations in the

actual geoid caused by subsurface inhomogeneities that invalidate the fluid earth

hypothesis.

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The mathematical description of the gravitational acceleration on the

reference ellipsoid incorporates the flattening information from satellite orbits, the

angular frequency, the equatorial value of g, and the equatorial radius in a closed

form expression for the value of g as a function of latitude on the reference spheroid.

A reference field, World Geodetic System 1984 (WGS84), is given by:

2

02

1 0.00193185138639sing 9.7803267714

1 0.00669437999013sin

(2.1.11)

For this reference spheroid:

The equatorial radius, Re, is 6378.137 km

The flattening parameter, f, is 1/298.257

Acceleration at the equator, ge, is 9.7803267714 ms-2

This reference field is used to remove the first order variations due to latitude that

are observed in gravity surveys. While not corresponding to the actual shape of the

earth the equipotential ellipsoid is in a way a mathematical construct that is used by

geodesists as a reference shape for surface elevation mapping and satellite

positioning. The gravity field of this reference ellipsoid fulfills the same role for

geophysicists interested in perturbations of the gravity field caused by subsurface

density variations.

The actual equipotential surface, the geoid, is usually described in terms of its

departure from the reference ellipsoid. Alternatively the values of g on the geoid

can be described with respect to the values of g for the reference ellipsoid. The

actual geoid is found to be a very complex shape with undulations in its surface with

a wide range of length scales. The only place where the geoid is easily ‘seen’ is the

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ocean surface that is the equipotential that defines the geoid. Over land the geoid is a

less intuitive concept. The usual concept is to imagine canals that extend through the

land mass. The water surface in these canals are sections of the geoid. The details of

the geoid were first glimpsed in fitting scattered worldwide absolute measurements

of g and these indicated a pear shape superimposed on the ideal ellipsoid. A much

more detailed geoid was derived from interpretation of satellite orbits.

The general form for the potential of an arbitrary spheroid is given by an

expansion of Laplace’s equation for the potential in spherical harmonics. The first

term in the expansion accounts for the flattening of the spheroid but higher order

terms account for distortions of gravity caused by density inhomogeneities or

deflections of the surface on any scale. The general solution for the Laplace equation

in spherical coordinates describes the potential above the earth. From the potentials

the accelerations are derived and in turn the forces acting on an orbiting satellite are

derived to determine the orbit. An inverse procedure is used to convert the observed

satellite orbits and their precessions and irregularities into estimates of the

coefficients of the terms in the solution to Laplace’s equation. With the solution

established, all the equipotentials, including the one that corresponds to the geoid,

and gravity values can be specified. Whether the gravitational field is described by

an equipotential or in a derivative of the potential – a gravitational acceleration – is a

matter of choice. They are of course directly related and in the case of a simple

isolated mass beneath the surface there is a simple relation between the geoid height

the potential and gz. Consider the model of a seamount shown in the sketch of Figure

2.1.6.

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Figure 2.1.6

The seamount is a volume of excess mass M’ with respect to the surrounding ocean.

The potential U above it must be bigger than at points off to the side. Consequently

one would have to move to a bigger earth radius to be at the same potential as found

off to the side. The equipotential surface is therefore warped up over a subsurface

body of excess mass. Since the ocean surface is an equipotential the ocean has an

induced surface ‘dome’ over the seamount.

In detail the background standard gravity gz away from the seamount is:

2

ez

e

GMg

R

and the potential is e

e

GM

R .

M '

Geoid

H

A erSmithandSandwell2004

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As we move to a point over the excess mass M’ the potential increases so the

background equipotential must move to a bigger earth radius, eR H so the

change in potential U must satisfy

1

1

e e e e

e e e e

e

GM GM GM GMU

R R H R R H

R

Since eH R we have:

21e e e

z

ee e e

GM GM GMHU H g H

RR R R

(2.1.12)

This final equation, zU g H , relating the geoid height perturbation to the

potential difference and gravity is known as Brun’s formula.

A perhaps more useful formula relating the gravity anomaly g to the geoid

deflection is given by Li and Gotze (2001). For a two dimensional anomaly with a

spatial wavelength and geoid height anomaly of H the relationship is:

52 9.8 10 H cmg mgals

cm

(2.1.13)

The resolution of the gravity maps is about 10 mgal so as Li and Gotze point out a

10 mgal anomaly reflects a geoid deflection of only 1.6 cm. Conversely we will find

later that direct mapping of the ocean geoid surface by satellite radar altimetry has a

resolution of about 1.0 cm and so has a resolution of 6 mgal – just what is claimed.

Satellite measurements of gravity

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The standard approach for determining the geoid is to obtain the gravity

values from the tracking of satellite orbits and combining these values with absolute

gravity values measured on the earth’s surface and then fitting these values to the

general solution in spherical harmonics up to whatever degree and order is dictated

by the noise level of the data. The actual process is somewhat iterative because more

and more satellite data can be averaged for better and better gravity maps which in

turn lead to better and better definitions of the geoid and indirectly to better

integration of land surface gravity values,

A world gravity map based on two decades of satellite gravity alone

referenced to the WGS84 ellipsoid is shown in Figure 2.1.7 and the geoid height,

again referenced to the WGS84 ellipsoid is shown in Figure 2.1.8. Obviously the

gross features of both maps are similar related as they are through exact forms of

Brun’s formula or the formula from Li and Gotze.

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Figure 2.1.7. Gravity model based on decades of tracking geodetic satellites

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Figure 2.1.8. Geoid height

Satellite orbits are affected by all the terms in the expansion but it turns out

that the accuracy possible in monitoring the orbits limited the expansion to the first

30 or so terms. This basically restricted the definition of the geoid to features that

have fairly large spatial extent, on the order of 100s to 1000s of kilometers.

Figure 2.1.9 is a map of the height of the geoid from the reference WGS84

reference ellipsoid based on satellite data and ground data using spherical harmonic

coefficients to degree and order of 360. This map shows the anomalies in

remarkable detail. It is these perturbations of gravity and the geoid that are of

interest to geophysicists. The map shows upper mantle and crustal features that had

never been seen before and it has opened a whole new window on understanding

large scale geological processes.

All of them reflect certain departures from equilibrium conditions in a model

that, in the long term at least, is supposed to be fluid-like or at least plastic. If the

mantle is indeed viscous then a mass concentration should sink reestablishing

equilibrium and removing the geoid anomaly. Similarly if the crust is rigid the

accumulation of mass in crustal land masses should deform the geoid. The maps of

Figure 2.1.8 and Figure 2.1.9 show no broad correlation with the continents or ocean

basins so these features are indeed in equilibrium with the underlying mantle. The

most notable features are the major depressions over the Indian ocean, over the

northeastern part of Canada, off the west coast of Mexico and in the Atlantic north

of the South American continent and, at higher resolution, in the western US. It is

likely that the Canadian anomaly is due to the fact that its location was the center of

North American glaciation and the weight of the ice depressed the surface, the ice

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melted relatively quickly and the crust is rebounding on a longer time scale. Other

large scale anomalies could be due to asymmetries in the core-mantle boundary or to

convection in the mantle. Convection could lead to positive geoid height anomalies

by the upward deflection of the surface above a mantle upwelling. An example of

the fine detail available form the latest geoid maps is the by now famous high in the

geoid over Yellowstone shown in Figure 2.1.10.

Figure 2.1.10

This feature is interpreted as due to the upwelling of deep mantle rock into the

crust causing the Yellowstone hotspot. Here the geoid height reflects a regional high

in topography; the density anomaly and topography are not in fluid equilibrium.

Note also the well-defined geoid low coincident with the Central Valley of

California. We will return to this feature later in a discussion of even higher

resolution gravity data that has recently become available from the GRACE satellite

project.

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Satellites have also been used to map shorter wavelength distortions in the

geoid over the ocean. The ocean surface is of course the geoid but accurate maps of

its surface elevation were not possible until the advent of radar altimetry from a

stable orbit satellite. The radar altimeter actually maps the undulations in the geoid

surface shown in the sketch of Figure 2.1.11.

Figure 2.1.11

Repeat passes of the satellite average out the tidal and wave motions and the

resulting surface can be mapped to centimeters. The resolution obtained is

determined mainly by the noise introduced by wave and tidal height variations. To

some extent these are reduced by repeated averaging from many satellite passes. In

terms of anomaly resolution the altimetry data essentially takes over from the

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gravity data for all the short wavelength features (but only over the ocean). The

geoid surface over the ocean directly maps ocean bathymetry in addition to features

caused by mass excesses beneath the ocean. A map of the geoid surface converted

to bathymetry is presented in Figure 2.1.12. A more detailed portion of the Gulf of

Mexico and the Carribean is shown in Figure 2.1.13. Evidently the resolution is on

the order of 10 km. Converted to gravity the resolution is on the order of 10 mgals.

Better picking of radar arrivals and better averaging methods have dramatically

improved the resolution so that today the resolution is ~3 mgal with spatial

resolution of 5 km. Depth resolution is much less than that obtained from shipboard

sonar but then there is not much possibility of covering the oceans of the world with

a pattern of courses with close spacing. The gravity accuracy similarly is less than

that that can be obtained from modern ship borne gravimeters: 0.2 mgal with spatial

resolution of 0.25 km. Figure 2.1.12 and Figure 2.1.13 are from NOAA.

GRACE - Gravity Recovery and Climate Experiment

The direct measurement of gravity can be achieved using two identical

satellites spaced apart a distance D along a common orbit as shown in the cartoon of

Figure 2.1.14.

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Figure 2.1.14

When the satellite pair passes over an anomalous excess mass M’ in the Earth each

satellite is subject to an anomalous gravitational acceleration. One component is the

vertical gz which would have a small perturbing effect on the whole orbit, but

another component acts along the line joining the two satellites. This component

adds a small acceleration in the orbit path first towards the mass, zero over it and

then in the opposite sense past the mass. The result is that the satellite speeds up

approaching the mass and then slows back down after it. For a pair of satellites each

experiences exactly the same speeding up and slowing down but because they pass

over the mass at different times there is a relative velocity difference between them

as the pair passes over the mass. Wolf (1969) suggested that a pair of satellites

spaced apart about 200-500 km and equipped with a Doppler shift system to

measure the inter satellite velocity would have an effective sensitivity of 1.0 mgal.

His elegant analysis was based on potential and kinetic energies of an exact orbit

and the important conclusion was that the only the velocity along the orbit absorbed

significant energy.

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The general idea of such a twin satellite direct measurement scheme was

taken up in the design of the Gravity Recovery and Climate Experiment, GRACE.

The designers of the Grace system realized that microwave ranging systems could

be employed to measure changes in distance between the satellite pair to less than

10 micrometers. Figure 2.1.15 provides a qualitative explanation of how the Grace

system works.

Figure 2.1.15

Consider the distance S travelled by satellite 1 with velocity v along an orbital

path over an excess mass. The mass has a horizontal component of gravitational

acceleration gx shown in the figure as the cross over anomaly – attraction towards

the point over the mass to the left of the mass and opposite beyond the mass to the

right. The distance travelled is given by 21

2xS vt g t . Initially the time distance

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plot is a straight line with slope 1/v but as the horizontal acceleration increases the

satellite speeds up and the time-distance curve flattens out until the acceleration

cross over is reached and it is slowed back down and follows the original 1/v slope.

Satellite 2 follows behind at a distance D but it crosses the starting point of the

above graph at a later time so its time distance plot is simply displaced up. With

respect to the path of the lead satellite it can be seen that the separation first

increases to a maximum D’ above the mass lessens to a minimum D” just past the

mass and reverts to the normal separation beyond there. Thus the separation plot

mimics the acceleration. For a two dimensional mass and the assumption of a flat

Earth and horizontal flight path we will see in a later section that a profile of gx can

be converted to a profile of gz or to the potential itself by simple Fourier transform

operations. The GRACE satellite pair traverses a full polar orbit 16 times a day and

does not co-rotate with the Earth, so that it sweeps a complete three-dimensional

surface of gx values. Fitting these values to a spherical harmonic expansion allows

the gx values to be converted to values of gz or to the potentials. The fact that the

satellite pair has been in continuous operation for over 10 years has allowed a

tremendous amount of averaging to yield gravity maps of unprecedented accuracy.

Perhaps more important the continuous mapping coupled with the high accuracy has

permitted the mapping of gravity changes with time. Some examples below show

how this latter capability has opened a whole new window for applied geophysics in

groundwater monitoring, ice melting, tectonic processes and ocean dynamics.

A summary of the GRACE project and a comprehensive list of scientific

papers on it and the results obtained from it and be found at the GRACE website.

The GRACE satellite pair was launched in 2002 from Russia and put into polar orbit

at a mean height of ~500 km. The satellites are virtually identical, 220 km apart and

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they complete an orbit in 95 minutes. The Global Positioning System tracks them

very accurately and they have high sensitivity on-board accelerometers to measure

accelerations not caused by gravity.

In the first 110 days a world gravity map was released, Figure 2.1.16, that

revealed detail at least an order of magnitude better than any previous map (compare

to Figure 2.1.7). With time and averaging the resolution improved dramatically until

the point that changes in gravity over a period of a year could readily be seen.

An example derived from gravity changes between 2011 and 2013 is a study

of groundwater depletion in the Central Valley of California. The gravity data were

first converted to a change of mass and then converted to an equivalent change of

ground water level. The levels are shown in the 3 panels of Figure 2.1.17. It can be

inferred that the resolution is now on the order of 10.0 km.

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Figure 2.1.17

As indicated on the figure caption the mass deficit revealed by the gravity changes

shows that 15 cubic kilometers (~4×1012

gallons or 12.3 million acre feet) of water

is evaporating every year. The gravity data has introduced factual numbers into

running debates on water usage in California.

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Another remarkable result is that the GRACE data have been used to track changes

in the mass of ice in the Greenland ice cap (Velicogna, 2009)

Figure 2.1.18 (source NASA website)

The rate of ice mass loss has increased in the eight years that measurements

have been made and the melt in western Greenland is now accelerating. The detail

is so good that it can be seen that ice mass is actually being added to portions of

Greenland while being subtracted around the edges.

In this section we have gone from some basic potential field theory through to

the gravity field of the Earth and its first-order ellipsoidal shape. We have

introduced the concept of the geoid and shown how accurate measurements of the

geoid shape, using satellite orbits and satellite altimetry over the ocean, reveal

distortions in the gravity field caused by very large scale geological processes in the

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crust and upper mantle. These features are of a much broader scale than those

revealed in more traditional gravity surveys conducted on land or on the ocean for

resource exploration and studies of crustal geology. Finally we have seen how the

GRACE system measures gravity with an accuracy and resolution that overlaps the

gravity maps made on large scale ground and marine conventional surveys. We now

focus on measurement and processing of the much higher resolution gravity data

obtained in conventional land and marine surveys.