Variation of Gravity over the Surface of the Earth

## Latitude Dependent variations

Two features of the earth's large-scale structure and dynamics affect our gravity observations, its shape and its rotation. To examine these effects, lets consider slicing the earth from the north to the south pole. Our slice will be perpendicular to the equator, and follow a line of constant longitude between the poles.

Shape - To a first-order approximation, the shape of the earth through this slice is elliptical, with the widest portion of the ellipse aligning with the equator. Isaac Newton first proposed this model for the earth’s shape in 1687. Newton based his assessment of the earth's shape on a set of observations provided to him by a friend who happened to be a navigator on a ship, named Richer. Richer observed that a pendulum clock that ran accurately in London, consistently lost 2 minutes a day near the equator. Newton used this observation to estimate the difference in the radius of the earth measured at the equator from that measured at one of the poles and came remarkably close to the currently accepted values.

Although the difference in earth radii measured at the poles and at the equator is only 22 km (this value represents a change in earth radius of only 0.3%), this, in conjunction with the earth's rotation, can produce a measurable change in the gravitational acceleration with latitude. Because this produces a spatially varying change in the gravitational acceleration, it is possible to confuse this change with a change produced by local geologic structure. Fortunately, it is a relatively simple matter to correct our gravitational observations for the change in acceleration produced by the earth's elliptical shape and rotation.

To first order*, the elliptical shape of the earth causes the gravitational acceleration to vary with latitude because the distance between the gravimeter and the earth's center varies with latitude. As discussed previously, the magnitude of the gravitational acceleration changes as one over the distance from the center of mass of the earth to the gravimeter squared. Thus, qualitatively we would expect the gravitational acceleration to be smaller at the equator than at the poles because the surface of the earth is farther from the earth's center at the equator than it is at the pole.

Rotation - In addition to shape, the fact that the earth is rotating also causes a change in the gravitational acceleration with latitude. This affect is related to the fact that our gravimeter is rotating with the earth as we make our gravity reading. Because the earth rotates on an axes passing through the poles at a rate of once a day, and because our gravimeter is resting on the earth as the reading is made, the gravity reading contains information related to the earth's rotation.

We know that if a body rotates, it experiences an outward directed force known as a centrifugal force. The size of this force is proportional to the distance from the axis of rotation and the rate at which the rotation is occurring. For our gravimeter located on the surface of the earth, the rate of rotation does not vary with position, but the distance between the rotational axis and the gravity meter does vary. The size of the centrifugal force is relatively large at the equator, and goes to zero at the poles. The direction this force acts is always away from the axis of rotation. Therefore, this force acts to reduce the gravitational acceleration we would observe at any point on the earth, from that which would be observed if the earth were not rotating.

## Variations due to Excess Mass

The free-air correction accounts for elevation differences between observation locations. Although observation locations may have differing elevations, these differences usually result from topographic changes along the earth's surface. Thus, unlike the motivation given for deriving the elevation correction, the reason the elevations of the observation points differ is because additional mass has been placed underneath the gravimeter in the form of topography. Therefore, in addition to the gravity readings differing at two stations because of elevation differences, the readings will also contain a difference because there is more mass below the reading taken at a higher elevation than there is taken at a lower elevation.

As a first-order correction for this additional mass, we will assume that the excess mass underneath the observation point at higher elevation, point B in the figure below, can be approximated by a slab of uniform density and thickness. Obviously, this description does not accurately describe the nature of the mass below point B. The topography is not of uniform thickness around point B and the density of the rocks probably varies with location. At this stage, however, we are only attempting to make a first-order correction. More detailed corrections will be considered next.

## Variation due to Changes in Elevation

Imagine two gravity readings taken at the same location, at the same time, with two perfect (no instrument drift and the readings contain no errors) gravimeters; one placed on the ground, the other place on top of a step ladder. Would the two instruments record the same gravitational acceleration?

No, the instrument placed on top of the step ladder would record a smaller gravitational acceleration than the one placed on the ground. Why? Remember that the size of the gravitational acceleration changes as the gravimeter changes distance from the center of the earth. In particular, the size of the gravitational acceleration varies as one over the distance squared between the gravimeter and the center of the earth. Therefore, the gravimeter located on top of the step ladder will record a smaller gravitational acceleration because it is positioned farther from the earth's center than the gravimeter resting on the ground.

Therefore, when interpreting data from our gravity survey, we need to make sure that we don't interpret spatial variations in gravitational acceleration that are related to elevation differences in our observation points as being due to subsurface geology. Clearly, to be able to separate these two effects, we are going to need to know the elevations at which our gravity observations are taken.

## Variations due to Nearby Topography

Although the slab correction described previously adequately describes the gravitational variations caused by gentle topographic variations (those that can be approximated by a slab), it does not adequately address the gravitational variations associated with extremes in topography near an observation point. Consider the gravitational acceleration observed at point B shown in the figure below.

In applying the slab correction to observation point B we remove the affect of the mass surrounded by the blue rectangle. Note, however, that in applying this correction in the presence of a valley to the left of point B we have accounted for too much mass because the valley actually contains no material. Thus, a small adjustment must be added back into our Bouguer corrected gravity to account for the mass that was removed as part of the valley and, therefore, actually didn't exist.

The mass associated with the nearby mountain is not included in our Bouguer correction. The presence of the mountain acts as an upward directed gravitational acceleration. Therefore, because the mountain is near our observation point, we observe a smaller gravitational acceleration directed downward than we would if the mountain were not there. Like the valley, we must add a small adjustment to our Bouguer corrected gravity to account for the mass of the mountain.

These small adjustments are referred to as Terrain Corrections. As noted above, Terrain Corrections are always positive in value. To compute these corrections, we are going to need to be able to estimate the mass of the mountain and the excess mass of the valley that was included in the Bouguer Corrections. These masses can be computed if we know the volume of each of these features and their average densities.

## Variations due to tidal effect:

Tidal Affect - Variations in gravity observations resulting from the attraction of the moon and sun and the distortion of the earth so produced.

Superimposed on instrument drift is another temporally varying component of gravity. Unlike instrument drift, which results from the temporally varying characteristics of the gravimeter, this component represents real changes in the gravitational acceleration. Unfortunately, these are changes that do not relate to local geology, and are hence a form of noise in our observations.

Just as the gravitational attraction of the sun and the moon distorts the shape of the ocean surface, it also distorts the shape of the earth. Because rocks yield to external forces much less readily than water, the amount the earth distorts under these external forces is far less than the amount the oceans distort. The size of the ocean tides, the name given to the distortion of the ocean caused by the sun and moon, is measured in terms of meters. The size of the solid earth tide, the name given to the distortion of the earth caused by the sun and moon, is measured in terms of centimeters.

This distortion of the solid earth produces measurable changes in the gravitational acceleration because as the shape of the earth changes, the distance of the gravimeter to the center of the earth changes (recall that gravitational acceleration is proportional to one over distance squared). The distortion of the earth varies from location to location, but it can be large enough to produce variations in gravitational acceleration as large as 0.2 mgals. This effect would easily overwhelm the example gravity anomaly described previously.

An example of the variation in gravitational acceleration observed at one location (Tulsa Oklahoma) is shown above**. These are raw observations that include both instrument drift (Notice how there is a general trend in increasing gravitational acceleration with increasing time) and tides (the cyclic variation in gravity with a period of oscillation of about 12 hours). In this case the amplitude of the tidal variation is about 0.15 mgals, and the amplitude of the drift appears to be about 0.12 mgals over two days.

Time dependant Variations

Temporal variations of the Earth's gravity field are caused by a variety of complex phenomena including lunar-solar tides, atmospheric and oceanic mass redistribution, variations in groundwater storage and snow cover/ice thickness, earthquakes, post-glacial rebound in the Earth's mantle, long-term mantle convection and core activities, and other geophysical phenomena. It is important to understand these variations because of the implications they have for understanding and monitoring global climatic and geophysical changes, Earth rotation, and synoptic sea level changes. There have been a number of geophysical studies of the response of the Earth to loading and the secular changes this induces in the gravity field which would provide important constraints on mantle viscosity and sea level if reliable independent satellite estimates of the changes in the gravity field were available. Redistribution of the Earth's mass will also cause changes in the location of its center of mass, which have been measured using Lageos.

Temporal variations in gravity caused by ocean and solid Earth tides are relatively well determined because they occur at well known astronomical frequencies, but non-tidal variations in gravity are more difficult to detect. Recent progress has been made in both measuring and modeling temporal gravity variations.

The GRACE and GOCE Missions

GRACE - Short for Gravity Recovery and Climate Experiment, is a NASA mission consisting of twin satellites that were launched on 17 Maech 2002. The satellites are in the same orbit around Earth, one about 220 kilometers (137 miles) in front of the other at an altitude of 460 kilometers (286 miles) above the Earth's surface. Together, they measure Earth's gravity field with a precision greater than any previous instrument.

GRACE maps the entire gravity field of Earth every 30 days. Changes in gravity over time can reveal important details about polar ice sheets, sea level, ocean currents, Earth's water cycle and the interior structure of the Earth.

In the Arctic, for example, GRACE has found that the ice sheet that covers most of Greenland is shrinking. Measurements of decreasing gravity over the ice sheet, thus indicating a decrease in the ice sheet's mass, showed a loss of about 150 billion tons of ice per year between 2003 and 2006. Melting ice sheets and glaciers contribute to rising sea level worldwide. The melting Greenland ice sheet contributes about 0.3 millimeters per year to a rising global sea level. GRACE continues its measurements to understand whether this rate of sea level rise is increasing or decreasing.

In Africa and Australia, decreases in gravity are evidence of drying river basins. GRACE not only measures changes in water above ground -- in rivers, lakes and reservoirs -- but also senses the amount of water stored in aquifers beneath the Earth's surface. This kind of information can enable better management of scarce water resources.

GRACE can also detect changes in Earth’s crust brought on by earthquakes, such as the 2004 tremor in the Indian Ocean that triggered the deadliest tsunami on record. The quake changed the density of the rock beneath the surrounding sea floor, which GRACE was able to see several months later as a change in gravity.

GOCE Gravity Field and Steady-State Ocena Circulation Explorer, was launched in March 2009 by the European Space Agency at an altitude of 260 Km.  It has now collected enormous data on the gravity of the earth with unrivalled precision. The probe has enough fuel to fly until the end of 2012, a doubling of its intended mission life.  GOCE monitors ocean currents through temporal gravity changes.  By combining the gravity data with information about sea-surface height gathered by other spacecraft, scientists are able to track the direction and speed of ocean currents.  Goce data will have many uses, probing hazardous volcanic regions and bringing new insight into ocean behaviour.  Data from the five-metre-long spacecraft will be crucial for understanding sea level changes, shifts in ice flows and how ocean currents – which are driven by gravity – respond as the planet warms over the next few decades.

On the basis of gravitational variation in the earth’s crust, GOCE has mapped the surface of the geoid.

To see an animation of the geoid as constructed using data from GOCE, click here.

"GRACE is taking a movie and GOCE is taking a high-resolution still", is the analogy used by Dr Michael Watkins, the Grace project scientist at the US space agency's Jet Propulsion Laboratory.

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