_grav-1b gravity method

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

[email protected]

Department of Geology and GeographyWest Virginia University

Morgantown, WV

Gravity M ethods I-contiued


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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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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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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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Does water flow downhill?


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The notion of downhill is associated with a surfacealong which the gravitational potential decreases


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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.


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Geoid height anomalies

Contours are in meters

140 meters uphill


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Aside from wind

generated surfacewaves and ocean scale

wind generated swells

Is the ocean surface a flat surface?


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Map of the ocean floor obtained from satellite radar observationsof ocean surface topography.

SeaSat


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Detailed map of a triple-junction on the floor of the IndianOcean derived from ocean surface topography


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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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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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The anomaly shown here is

only 1/2 milligal

Karst


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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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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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( )2hR

mGg

E

Eh

+=

hsl ggg =

Is there another way to compute the change in g?


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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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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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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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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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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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Consider the following: what is the gravitationalattraction of a buried spherically symmetrical object?

Lets work through this on the board


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What is the vertical component?

cosRV gg =

( )1/ 2

2 2

cosz

x z =

+


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A symmetrical Earth holds no riddles for the geophysicist.


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If the earth were this simple our study would be complete.


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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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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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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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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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What other effects do we need to consider?

Latitude effect

Centrifugal acceleration

463 meters/sec

~1000 mph


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Solar and Lunar tides

Instrument drift


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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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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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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=


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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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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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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

_grav-1b gravity method

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