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8/23/2019 _Grav-1B gravity method
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Tom Wilson, Department of Geology and Geography
Environmental and Exploration Geophysics I
tom.h.wilson
tom.wilson@mail.wvu.edu
Department of Geology and GeographyWest Virginia University
Morgantown, WV
Gravity M ethods I-contiued
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Tom Wilson, Department of Geology and Geography
Rp = 6356.75km
RE= 6378.14km
gP=9.83218 m/s2
gE=9.780319 m/s2
This is a difference of 5186 milligals.
These kinds of differences, which in this case are a functionof latitude need to be corrected for or eliminated
2
EE
E
mg GR
=
Substitute for the
different values of R
There are great differences in the acceleration due to gravity on the Earth
that, in may instances, are unrelated to the details of subsurface geology
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Tom Wilson, Department of Geology and Geography
Density differences arising from isostatic equilibriumprocesses represent large scale regional changes of g that
are often removed before modeling and interpretation.
R. J. Lillie, 1999
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Tom Wilson, Department of Geology and Geography
Isostatic compensation and densitydistributions in the earths crust
R. J. Lillie, 1999
Generally geological processes
produce linear sheet like
distributions of materials
Its generally easier toaccept this kind of model
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Tom Wilson, Department of Geology and Geography
Does water flow downhill?
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Tom Wilson, Department of Geology and Geography
The notion of downhill is associated with a surfacealong which the gravitational potential decreases
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Tom Wilson, Department of Geology and Geography
The geoid is a surface of constant gravitational
potential. The gradient of the potential is perpendicularto the surface. Thus gravitational acceleration isalways normal to the equipotential surface.
8/23/2019 _Grav-1B gravity method
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Tom Wilson, Department of Geology and Geography
Geoid height anomalies
Contours are in meters
140 meters uphill
8/23/2019 _Grav-1B gravity method
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Tom Wilson, Department of Geology and Geography
Aside from wind
generated surfacewaves and ocean scale
wind generated swells
Is the ocean surface a flat surface?
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Tom Wilson, Department of Geology and Geography
Map of the ocean floor obtained from satellite radar observationsof ocean surface topography.
SeaSat
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Tom Wilson, Department of Geology and Geography
Detailed map of a triple-junction on the floor of the IndianOcean derived from ocean surface topography
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Tom Wilson, Department of Geology and Geography
In the environmental applications of gravitymethods anomalies smaller than a milligal canbe of interest to the geophysicist. A moderngravimeter is capable of measuring gravity to an
accuracy of about 100th of a milligal or better.Well spend considerable time discussing theapplications of gravity data in groundwaterexploration. An example of this application is
discussed in Stewarts paper (see web sitelink) on the use of gravity methods for mappingout buried glacial Valleys in Wisconsin - so readover this paper as soon as you can.
Gravity provides interesting views of objects buried
deep beneath the surface - out of our reach
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Tom Wilson, Department of Geology and Geography
Form Stewart
Bedrock models derivedfrom gravity dataResidual gravity data
The gravity anomalies associated with theseglacial valleys have a range of about 4 milliGals.
Why residual? The residual eliminates the influence of the
deeper strata which dip uniformly across the area. Their
configuration is not relevant to the problem at hand. The
residual can eliminate geology we arent interested in
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Tom Wilson, Department of Geology and Geography
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Tom Wilson, Department of Geology and Geography
The anomaly shown here is
only 1/2 milligal
Karst
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Tom Wilson, Department of Geology and Geography
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Tom Wilson, Department of Geology and Geography
These variations in gravitational acceleration are verysmall. To give you some additional perspective on themagnitude of these changes, consider the changes ing as a function of r (or RE) as indicated by Newtonslaw of gravity -
2
E
E
R
mGg =
Recognize that the above equation quantifies thevariation in g as a function of r for objects that can
effectively be considered as points. For now, lets take
a leap of faith and assume that we can represent theEarth as a point and that the above equation accuratelydescribes the variations in g as a function of distance
from the center of the earth, RE.
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Tom Wilson, Department of Geology and Geography
2E
Esl
R
mGg =
Given this relationship -
RE
hWhat is g at a distance
RE+h from the center ofthe earth?
sl=sea level
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Tom Wilson, Department of Geology and Geography
( )2hR
mGg
E
Eh
+=
hsl ggg =
Is there another way to compute the change in g?
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Tom Wilson, Department of Geology and Geography
2E
E
R
mGg =What is the derivative of g
with respect to R?
=
2E
E
R
mG
dR
d
dR
dg
( )2= EE RdR
dGm
dR
dg
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Tom Wilson, Department of Geology and Geography
At Morgantown latitudes, the variation of g withelevation is approximately 0.3086 milligals/morapproximately 0.09406 milligals/foot.
As you might expect, knowing and correcting for
elevation differences between gravity observationpoints is critical to the interpretation and modeling ofgravity data.
The anomalies associated with the karst collapse
feature were of the order of 1/2 milligal so an error inelevation of 2 meters would yield a difference in ggreater than that associated with the density contrastsaround the collapsed area.
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Tom Wilson, Department of Geology and Geography
J ust as a footnote, Newton had to develop the mathematicalmethods of calculus to show that spherically symmetrical objectsgravitate as though all their mass is concentrated at their center.
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Tom Wilson, Department of Geology and Geography
The acceleration term in Newtons law of gravitation.
2r
mGg =
tells us we need to consider mass (m) and itsdistance(s) (ri) from some observation point. In practicewe usually compute the acceleration of some arbitrarily
shaped mass by breaking it up into small parts andsumming their individual contributions to g.
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Tom Wilson, Department of Geology and Geography
2, , or L S V
Gdmg
r=
Integral form of Newtons law of gravitation
Line, surface or volume
Depending on symmetry
2
G dVg r
=
2
G dxdydzg
r
=
dz
dy
dx
dV
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Tom Wilson, Department of Geology and Geography
Consider the following: what is the gravitationalattraction of a buried spherically symmetrical object?
Lets work through this on the board
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Tom Wilson, Department of Geology and Geography
What is the vertical component?
cosRV gg =
( )1/ 2
2 2
cosz
x z =
+
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Tom Wilson, Department of Geology and Geography
A symmetrical Earth holds no riddles for the geophysicist.
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Tom Wilson, Department of Geology and Geography
If the earth were this simple our study would be complete.
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Tom Wilson, Department of Geology and Geography
How thick is the landfill?
Gravity methods thrive on heterogeneity. In general the objects weare interested in are not so symmetrical and provide us with
considerable lateral density contrast and thus gravity anomalies.
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Tom Wilson, Department of Geology and Geography
How does g vary from A to E?
We might expect that the average density of materials in the landfill wouldbe less than that of the surrounding bedrock and thus be an area of lower g
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Tom Wilson, Department of Geology and Geography
At present weve only accounted for variations in g as a function of elevation ordistance from the center of the earth. But obviously we have further to go in termsof conceptualizing and developing the computations needed to understand and
evaluate geological problems using measured gravitational fields.
Another variable for us to consider is theelevation at which our observations are made.
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Tom Wilson, Department of Geology and Geography
How do we compensate for the influence of
matter between the observation point (A) and sealevel?
How do we compensate for the irregularities inthe earths surface - its topography?
A hill will take us down the gravity ladder, but as we walkuphill, the mass beneath our feet adds to g.
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Tom Wilson, Department of Geology and Geography
What other effects do we need to consider?
Latitude effect
Centrifugal acceleration
463 meters/sec
~1000 mph
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Tom Wilson, Department of Geology and Geography
Solar and Lunar tides
Instrument drift
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Tom Wilson, Department of Geology and Geography
To conceptualize the dependence ofgravitational acceleration on variousfactors, we usually write g as a sum ofdifferent influences or contributions.These are -
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Tom Wilson, Department of Geology and Geography
gn the normal gravity of the gravitationalacceleration on the reference ellipsoid
gFA the elevation or free air effectgB the Bouguer plate effect or the contribution tomeasured or observed g of the material between sea-level and the elevation of the observation point
gTthe effect of terrain on the observed ggTide and Drift the effects of tide and drift (oftencombined)
These different terms can be combined into anexpression which is equivalent to a prediction of whatthe acceleration should be at a particular point onthe surface of a homogeneous earth.
Terms
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Tom Wilson, Department of Geology and Geography
Thus when all these factors arecompensated for, or accounted for, the
remaining anomaly is associated with
lateral density contrasts within area of the
survey.
The geologist/geophysicist is then left with thetask of interpreting/modeling the anomaly in
terms of geologically reasonable configurations ofsubsurface intervals.
The gravity anomaly obs t g g=
8/23/2019 _Grav-1B gravity method
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Tom Wilson, Department of Geology and Geography
That predicted or estimated value of g is often referred toas the theoretical gravity - gt
If the observed values of g behave according to thisideal model then there is no geology! - i.e. there is nolateral heterogeneity. The geology would be fairlyuninteresting - a layer cake ...
Well spend more time with these ideas, but in the nextcouple lectures we will develop a little betterunderstanding of the individual terms in this expression.
( )t n FA B t Tide Drift g g g g g g += +
The Theoretical Gravity
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Tom Wilson, Department of Geology and Geography
Well carry on this discussion in greater detail nexttime. Make sure you continue reading chapter 6 inBurger et al.
Well go over some of the basic ideas associated
with the plate correction and the topographic (orterrain) correction.
The basis for these two corrections are associatedwith the gravitational acceleration produced by a
plate of finite thickness but infinite horizontalextent and by individual sectors from a ring ofgiven thickness and width.
Read general introduction from pages 349-355 and continue
reading about gravity corrections through the top of page 373
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Tom Wilson, Department of Geology and Geography
Keep reading Chapter 6.
Hand in the three intro gravity problems
before leaving today
Gravity papers are in the mail room!
Start looking over problems 6.1 through 6.3
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