Eart 110N The Dynamic Earth Lab
Lab#6—Earth’s Gravity
Objectives: You will measure the rate of decrease of gravity with height in Baskin Engineering
Building, using a Lacoste-Romberg gravimeter, and see how well it compares with the standard,
calculated free air correction. You will then convert your relative gravity readings to absolute
gravity values by comparison with the known value at a base station, find the free-air gravity
anomaly for each of your gravity stations and, finally, calculate the Bouguer anomaly.
Introduction: The absolute acceleration of gravity, g, as opposed to the universal gravitational
constant, G, can be measured with considerable accuracy. One needs only to time the period of a
precisely constructed pendulum or the duration of free fall of an object in a vacuum. Changes in
gravity from one place to another can be measured even more accurately—to within one part in
108 or 109—with a gravimeter. By tying these relative changes to stations where absolute g is
known, geophysicists made gravity maps of continental areas during the first half of the 20th
century that revealed lateral variations in density of material below the surface. A Dutch
geophysicist, Venning Meinesz, also made less accurate gravity maps of some oceanic areas by
means of compensating (counter-swinging) pendulums mounted in a submarine. Now global
gravity maps are made using satellites: by tracking perturbations in the orbits of satellites and by
measuring small topographic variations in the sea surface.
Earth’s gravity departs from that of a spherically symmetric body, GM/r2, because of its rotation.
The centrifugal force causes the earth to bow out at the equator and in at the poles, giving it a
spheroidal shape. This simple shape plus the direct effect of the centrifugal force dictate the
average latitudinal variation of gravity:
g() = geq(1 + sin2 + sin4)
where  is latitude and
(1)
geq = 978031.846 mgal (1 mgal = 10-3 cm/sec2 = 10-5 m/sec2)
 = 5.302359x10-3
 = 5.8655x10-6
The value of g given by equation 1 is called gravity on the spheroid or standard gravity. These
values of the numerical coefficients were adopted by the International Union of Geodesy and
Geophysics in 1967 as providing the best fit of this smoothly varying function of latitude to the
actual global gravity measurements at the time. Slightly better values could now be calculated,
but the IUGG 1967 standard gravity formula is still retained in calculations of gravity anomalies
for the sake of uniformity.
The spheroid is the shape the earth would have if, while maintaining the same mass, lateral
density variations along each thin spheroidal shell within the earth were smoothed away and a
(shallower) ocean of uniform depth covered the surface. This shape is the reference surface for
maps. Gravity anomalies are defined relative to that expected on the spheroid. For stations at sea
level the gravity anomaly, which is always defined as the observed minus the expected value, is
g  gobs  g(), because gexp is gravity on the spheroid—i.e., g() from equation 1.
For stations above sea level, gexp is smaller than g() because the decrease in gravity with
increasing distance above the spheroid is easily calculated and quickly swamps the variation in
gravity due to lateral density variations inside the earth that we are most interested in. For the
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elevation range found on earth this variation in gexp with elevation is, to a satisfactory
approximation, the same as for a perfectly spherical earth, and is called the free air correction
FA_corr. It is given by
FA_corr 
GM earth
g
d GM earth
(
) Δh  (2
) Δh  2 Δh  0.3086 Δh mgal
2
3
dr
r
r
r
(2)
where g is the average acceleration of gravity and h is elevation in meters.
Then the free air anomaly is
gFA  gobs  gexp = gobs  (g() + FA_corr)= gobs  (g()  0.3086h)
(3)
Its peculiar name stems from the notion that it only corrects for elevation above (or below) the
spheroid, as though there were nothing but free air between the observation point and spheroid.
On land, the Bouguer anomaly is a refinement of the free air anomaly that corrects
approximately for the attraction of mass that lies above the spheroid. As shown in a lecture
handout, the simple Bouguer correction is the attraction of a planar horizontal layer of infinite
lateral extent and thickness h and is given by
SB_corr = 2Gh = 0.1119h mgal
(4)
where h is elevation in meters and the density contrast is  = rock  air  rock  2670 kg/m3.
At sea the simple Bouguer correction is taken to be
SB_corr = 2Gd =  0.0675d mgal
(5)
where d is ocean depth in meters and the density contrast is  = seawater  rock  1610 kg/m3.
The simple Bouguer anomaly is
gSB  gobs  gexp = gobs  (g() + FA_corr +SB_corr)
(6)
Gravity Meter: The Lacoste-Romberg gravimeter is basically a mass suspended on a spring. It
measures changes in gravity by measuring the amount the spring stretches or contracts. The
instrument is insulated and connected to a battery and heater that keep it at a constant
temperature. It is very sensitive and extremely delicate when the suspension is hanging free.
Always keep the suspension clamped except when you are taking a reading!!
Gravity Measurement Procedure: The procedure for each measurement will be:
1. Keeping the suspension clamped and the connection to the battery inside the case intact,
remove the gravimeter, place it on the hemispherical support, turn on the light and level the
instrument (exactly!). Exact leveling will be difficult or impossible if people are moving near the
gravimeter.
2. Unclamp the instrument gently and completely (turn knob fully anticlockwise) and turn the
big aluminum wheel left or right until the black pointer on the little round dial comes on scale.
Turn the wheel so the pointer is well to the left of middle on the dial. Check and fine-tune level.
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3. Look through the eyepiece and slowly turn the aluminum wheel toward larger values. The
indicator line should be on the left side and moving slowly toward the middle. Check and finetune the level. Very slowly turn the aluminum wheel in the same direction until the left edge of
the indicator line lines up with the 3.10 reading line. By always approaching the reading from the
left side, you avoid possible backlash in the screw system and obtain more reproducible results.
4. Check the level one more time. If it is off, level the instrument again, back off the wheel so
that the pointer is to the left of middle, and repeat step 3. If it is good, gently clamp the
instrument all the way clockwise and turn off the light.
5. Record the time as well as the gravimeter reading. For example, if the digital dial reads 34054
and the aluminum wheel reads 446, the reading you should record is 3405.446—that is, the last
digit on the dial should be the same as the first number on the aluminum wheel. The third
number read from the wheel (6 in this example) will be very rough at best.
6. To make replicate measurements, repeat steps 2-5.
7. Carefully put the gravimeter back in the case, close the case up, and move to the next station..
Gravity Measurements: We will make our measurements at the Baskin Engineering Building to
take advantage of a secondary base station where we know the value of absolute gravity.
a. Station 1: Base station at circular paint spot on the concrete walkway just outside the stairway
at the SW corner of the building (near the post office). Absolute g = 979900.55 mgal, elevation
h = 243.2  1.5 m, and latitude  = 37.000 N.
b. Station 2: Ground floor inside the stairwell at the small white “X” marked on the floor.
c. Station 3: First floor inside the stairwell at the small white “X” marked on the floor.
d. Station 4: Second floor inside the stairwell at the small white “X” marked on the floor.
e. Station 5: Third floor inside the stairwell at the small white “X” marked on the floor.
f. Station 1: Base station again. Your two measurements of gravity and time will enable you to
assess and correct for instrument drift, which you should assume occurred linearly with time.
Elevation Measurements: Use the tape to measure the heights of the stations to the nearest
centimeter above the base station. You will need to use a level to measure the height of station 2
above station 1. For the rest of the stations you can dangle the tape down the stairwell. Devise an
accurate way to measure the elevation difference between stations 4 and 5, which is complicated
by the presence of a concrete abutment that is in the way.
Analysis: After you make the measurements, you should return to the lab and start your analysis
of the data, which gets a little involved. You should try to finish steps 1-2 below and at least
begin on steps 3-4, so you understand how to turn your readings into absolute milligals. At home
you can enter your data into spreadsheet rows for each station and complete your analysis and
discussion.
1. Each gravimeter manufactured is calibrated against a known g change when it is
manufactured or repaired, and each has a different set of multiplicative constants to use to adjust
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the relative gravity differences to milligals. In our case, assuming the readings are between 3400
and 3500, convert your reading to milligals using
Adjusted Value = 3596.98 + 1.06096(Gravimeter Reading  3400)
(7)
2. Use base-station readings a and f to estimate the drift rate. Then correct the individual
readings for drift by using the time each of them was made. If replicate measurements were
made, find the mean and standard deviation for the drift-corrected readings at each station.
3. Enter your relative elevations, times, and drift-corrected gravimeter readings for each station
in Excel, and use equation 7 to adjust them to relative milligals. These ‘adjusted’ g values
should be placed in the column next to the drift-corrected raw values.
4. Enter the value of absolute gravity for the base station in the next cell to the right of its
adjusted gravimeter reading. Use this to convert your relative gravity readings at stations 2
through 5 to absolute gravity values in milligals and enter them in the same column containing
the adjusted value for station 1. Using the base station elevation above sea level, figure and enter
elevations above sea level for stations 2 through 5 and place them in the next column to the right.
5. Plot your absolute gravity values versus elevation above sea level in Excel. Show the bestfitting straight line to your points, the equation of this line, and the R2 value describing the
goodness of fit. To do this, select your graph rectangle, use menu commands Chart/Add
Trendline/Options and choose ‘Linear’ to show the best-fit line, and check the boxes to display
the equation and R-squared value on the chart. Select the panel on your chart containing the
equation of the line and R2 and, using menu commands Format/Selected Data Labels/Number,
click ‘Number’ and dial in 4 decimal places.
6. How well does the coefficient of x in the equation of your best-fit line agree with the
calculated coefficient of h in the free air correction (equation 2)? Calculate the percent
difference between the two.
7. Evaluate whether the difference could be entirely due to measurement uncertainties. In other
words, estimate the percent errors in your measurement of gravity and height differences
between the ground and third floors. Are they big enough to account for the discrepancy in the
coefficients in number 6 above? Or is it necessary to ascribe the discrepancy to some real
differences between the real earth and the simplified earth model assumed in equation 2?
8. Use equation 3 and the latitude to calculate* the free air anomaly for stations 1 through 5,
placing these values in the next column to the right.
9. Use equation 6 to calculate* the Bouguer anomaly for the station 1.
10. Generally, Bouguer anomalies are large and negative (e.g., ~ 100 mgal) over continents and
large and positive (e.g., ~ +100 mgal) over ocean basins, whereas free air anomalies are small on
average over both continents and oceans (e.g.,  10 to 30 mgal). How typical is our base station?
Any ideas to explain the abnormalities? (P.S. I don’t have the answer to this one myself.)
* Be careful to keep track of the signs of the corrections in your calculations!
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