# Burger Ch06 Gravity

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Chap 6 - Gravity
on the face of it, more straightforward than resistivity

you measure gravity at the surface of the earth, interpret the values

need very sensitive equipment
corrections must be applied
elevation (further up from sea level the less the pull)
remove effects of nearby bodies

still left with problem of multiple configurations giving the same gravity

I. Fundamental relationships
A. gravitational acceleration

F = G m 1 m2
r2
which also can be written as:

g = GM
R2
where G = grav constant, M = mass of Earth, R = radius of Earth
II. Measuring gravity
A. relative msrmt discussed first

1. using a pendulum
period of an ideal pendulum represented by

T = 2 K/g

imprecise, but way we get around it is to make 2 msrmts at 2 diff points -
we then can at least get the relative difference between gravity at the 2
stations

g = g obsy - g obsx

2. Using a gravimeter
classic unit - the Worden gravimeter - still rely on relative diffs between 2
stations…but now the precision is 1 in 100 million, or 0.01 mGal

corrections need to be made for drift
B. absolute measurements
falling body approach, relies on amt of time it takes body to fall

z = gt2
2

III. Adjusting the observed gravity values
need to determine how grav varies…

•variance in position on Earth’s surface latitudinally causes diffs in grav...
- due to rotation of Earth, centrifugal force acts outward strongly at
Equator, less at poles, so g is less at Equator by 3.4 Gal
- due to flattened poles, g stronger by 6.6 Gal at the pole
- less mass between pole and center of earth vs equator and center
of Earth causes g to be less strong at pole by 4.8 Gal

somehow all factors add up to be 978 gal at Eq, 983.2 at pole

A. so…correct for latitiude
B. correct for elevation:
•free air
•Bouguer (“boo-gay”, not “booger” like something out of your nose…)

1. Free air - get the first derivative of the gravity equation,
dg = - g 2 = -0.3086 mGal
dz        r       meter

that is, grav decreases by .31 mgal for each meter above the reference
plane (sea level) DUE to ELEVATION ALONE
above sel level, subtract it when below sea level.

2. Bouguer correction - this one is due to mass that is between
observation pt and sea level…why do we need to correct for this?
Fig 6-3 Because you need to compare what is under A and B

3. Terrain correction - Fig 6-5 - you may need to re-correct for the
Bouguer correction, since it’s simplistic

this is time-consuming, complicated
B. Isostasy and anomalies - principle is like icebergs - low density
mountains with deep roots, floating in a sea of high density oceanic
material

Mountain
Ocean
IV. Field procedures - things to consider…
A. Drift and Tidal effects - Fig 6-8 shows typical drift

drift due to 1) instrument changes, 2) tidal effects due to position of Sun
and Moon

1) “looping” with gravimeter brings you back to your base station
draw a line showing the drift through time at base, and a bunch of
parallel lines running through the real data points.

values…

2) the tidal correction curve is more complex..computer is useful for
correction…must back out the tidal correction and the drift correction in
order to have usable gravity values

B. Establish base stations where you know absolute gravity if possible
(use an IGSN71 locale…)..this helps you “root” your relative values.
C. Determining elevations
this shows the progression of technology in the past few years -
•surveys originally by rod and transit, tied into USGS benchmarks
(which were also surveyed in from other benchmarks); these are now
accurate laser-based devices known as total stations (we’re trying to
get one with the Dean’s \$)
•spot elevations are those to +/- one foot, usu located on topo maps
•in urban settings, much of the data may be already available through
city planners & engineers
•reference to GPS as the new alternative to rod and transit surveys,
“will become the dominant approach in the not too distant future”
future is now - research proposal just submitted requesting
\$5000 for GPS unit accurate to .1 meter (4 “).
D. Determine horiz positions - authors show how to get accurate to .01
minutes of latitude, but this is task excellent for GPS (GPS better for
horiz positioning than vertical, just because of satellite geometry)
E. Bouguer reduction density - discussion of what value to use - standard is
2.67 g/cc. Recall that Bouguer correction is the one where we remove the
mass between the observation point and sea level.
F. Survey procedure - example from the Smith College class
Table 6-4 shows all the data collected

V. gravity effects of simple shapes (fun stuff)
we’ve learned how to correct the data
once we make the maps we’ll want to interpret the anomalies and relate
them to subsurface geology and density anomalies.
So one logical step is to determine what some basic shapes due in terms of
creating gravity anomalies.
•First, discuss rock densities - get bulk density (of saturated material) by
adding the proportion of solid phase with the proportion of the liquid phase

bulk = min 1-poro) + H2O  poro 
 100            100 
B. gravity due to a sphere
simple so this is good to start with..analog is an equidimensional ore
body
derive an equation for grav attraction on the surface due to the body

g z sphere = G 4  R3 c   cos 
3 (x2 + z2)               where c = density contrast

Fig 6-14 shows the curves for a shallow sphere and deep sphere
- curve centered and strongest over the spheres, tails off on
either side
- bigger the dens contrast or shallower the body, the stronger
the anomaly
can use this shape characteristic to make estimates about depth to the
anomaly

can use Table 6-6 and vary parameters, graph the results
C. grav effect of a horizontal cylinder

now we can pretend that the sphere’s 2D gravity profile is just
stretched out along the long axis of the cylinder...

z

x

The 2D equation, in the x-z plane, is
g z cylinder = G 2  R3 c cos 
(x2 + z2)
note the similarity between the sphere and the cylinder (duh…!)
D. gravity effect of a vertical cylinder
also a useful model device…

g vert cylinder = G 2  c   h2-h1 = R2 + h12 -  R2 + h22 )

however, note that this is good only for the pt above the axis!

E. grav effect of inclined rod (like a dipping ore body? Plunging anticline?)

F. grav effect of inclined sheet….essentially the areal distribution of a
bunch of thin rods

leads to some interpretation with faults….Fig 6-21

on to using the computer program - GravModel

next phase of analysis
VI. Analyzing anomalies

authors note a diff between grav and the seis & resis methods worked
earlier

we don’t necessarily know as much about where our sources are when
it comes to grav…we want info from shallow, but we may see info from
deep as well.

Thus we need to figure out how to remove large trends, in order to view
the fine detail…

A. Regionals and residuals

regional trend removal leads to ID of smaller anomalies (“residuals”)

so how get the regionals? Take profiles across a contoured map (profile
is a x-sec, that’s all…), look for the main, smooth trends

then subtract the first map from the trend map, and you should see
individual anomalies
Real data
g

Regional trend

West                                                   East

What are you left with?
Fig 6-23 - real world example - river channel in sed, over basement
actually, what they do here is construct a geol model and then make a
geophys curve from the model
6-24a shows the real data, b shows the regional curve that is essentially
due to bedrock and sed, not considering the less dense river channel
b also shows the residual, which is the curve that you get when you
subtract the regional from the actual
note the depression in the residual grav curve

B. Trend Surfaces
mathematical surfaces - “best fit” - we saw these in GIS class
minus to this approach is that there’s no room for geology - all math
plus to approach - no bias!
look at 1st, 3rd, 5th order polynomials…fig 6-26
point here is that the higher order the poly, the less easy it is to see the
anomaly - realworld examples shown in 6-27, with the trend surface map
in 6-28
Tough to visualize without using a profile…

C. Upward/downward continuation - talk about potential field theory a bit
if you measure the geology from higher elevations, you mask the shallow
anomalies…they are overwhelmed by the regional gravity
D. 2nd derivatives - here, you look at the rate of change of slope
(the rate of change of the rate of change…)
this means where the curvature of the gravity field (or any map) is
greatest is where your 2nd derivative values are also the greatest
often used for detecting very small, subtle changes
E. filtering briefly discussed - can filter gravity data just like we do seis
data…pull out the long wavelengths in order to see the short ones

VII Grav interpretation
suppose we’ve taken the data, removed the regionals…we’re ready to
interpret
typical approach, like electrical methods - put a model in, see how clse
you get to reality
Caution here, just like resistivity - you tweak the model the best you can to
get it to fit the data, but that is NO GUARANTEE that it is the TRUE
geology
but we do have some techniques that help us narrow the possibilities a bit
1.half-maximum - derivation of technique almost understandable (!)
we are given that the depth to the center of a sphere, z is:
z = 1.305 x 1/2 max
so if you measure the horiz distance from the crest of anomaly to the 1/2
value point, and multiply that no. by 1.305, you should have a guesstimate
as to the depth to the center of the object you are modeling as a sphere
COOL!!
For a cylinder, relation is

z = x 1/2 max

Authors note that other “half-max” expressions exist too….
2. 2nd deriv techniques
Method of Stanley (1977) shown as example of technique - can
convert 2nd deriv values of max and min to depth to top of body and
dip of lateral contact

IX. Applications of Gravity Method - Fig 6-36
A. Find bedrock depth
case history - urban setting, trying to site a tunnel in bedrock

B. find subsurface voids- Fig 6-37
setting - sinkhole terrain in Marion Co, FL.
Survey picked up both irregular topography as well as voids at depth
authors also note the usefulness resis plus the grav

C. Landfill geometry Fig 6-38 and Fig 6-39
note the correspondence between landfill thickness and the negative
gravity…..

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