Potential Field Methods

8/9/2019 Potential Field Methods

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Potential Field Methods

• + natural source methods • + non-invasive • + inexpensive • + fast • + easy data collection, reduction, but... • - non-straightforward interpretation • - low resolution • - ambiguous • - not always applicable

Method Advantages Disadvantages Cost Ratio

Magnetics Very fast, very cheap Poor resolution, not always

applicable 1

Gravity Fast, cheap Poor resolution 10

Seismic Fine detail, good correlation to

geology $$$ 100

From a 1999 Edcon brochure advertising their aerogravity/magnetic surveys:

"The cost of conducting an aerogravity/magnetic survey over a 5,000 square kilometerconcession in South America is in the order of $200,000 to $300,000. The cost of a 3-Dseismic survey over only 250 square kilometers can be ten times that amount."

Pat Millegan, Marathon Oil, on use of G&M in industry:

Pat stresses the importance of diversifying your skills: " seismic does NOT answer allthe questions, all the time...there are MANY seismic failures (e.g., one current Marathon project). The main reason G&M does not see more use is true "ignorance". My job is 10-100 times harder when my "clients" (the exploration groups...I'm in a service group)know nothing about G&M. Please stress geophysical integration to your students. It isthe smart way to explore, but you don't just throw G&M at everything...don't bother if thegeology isn't conducive to geophysical results."

1972 Costs of Acquisition and Processing of Geophysical Data (Telford etal.)

x $106 %

Petroleum Exploration

seismic 802 89.7

surface grav/mag 17 1.9

airborne mag 6 0.7


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Mineral Exploration

airborne mag 19 2.1

ground mag 12 1.5

Other 34 3.8

Total 894 100

Gravity and Magnetics in a Nutshell

Gravity is useful wherever the formations of interest have densities that areappreciably different from those of surrounding formations.

Some examples:

• mapping sedimentary basins, where sedimentary rocks consistently havelower density than basement rocks

• salt bodies: low density of salt • groundwater studies (e.g., Cayman Islands) • burial chambers in pyramids

Magnetics is useful whenever object of investigation has a contrast in magneticsusceptibility or remanence

Some examples:

• mapping structure on basement • mapping sedimentary basins • direct location of ores containing magnetite

History of Gravity Method

• Man has always recognized its force: fear of falling; up & down • Galileo, 1590: pendulum period; force on body proportional to weight;

acceleration of g independent of mass; gal = 1 cm/s2 • After sun recognized as center of universe, Tycho Brahe (1546-1601)

made extensive measurements of the "peculiar motion" of planets • Johannes Kepler (1571-1630): Kepler's Laws history

1. The planets move in elliptical orbits with the sun at one focus


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where a (distance CA below) and b (distance CB below) are the major andminor semiaxes. The eccentricity, e, is given by c/a, where c is thedistance from the center of the ellipse to one of the foci, and x and yrepresent coordinates of points on orbit. (examples: Earth = 0.01673;Mercury = 0.2056; Pluto = 0.250)

1. A line drawn from the sun to a planet will sweep out equal areas in equaltimes (conservation of angular momentum)

2. The square of a planet's period of revolution is proportional to the cube ofthe length of the major semiaxis of the orbital ellipse (conservation ofkinetic and potential energy)

• Newton, 1687, Philosophiae Naturalis Principia Mathematica : force ofgravity is a property of all matter, Earth included

• Jean Richer, 1672: pendulum clock, accurate in Paris, lost a few minutesper day in Cayenne, French Guiana


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Seen as tool to measure variation in geopotential. Newton correctly interpreted as due tooblateness. French believed otherwise; French Academy of Sciences sent two expeditions, oneto high latitudes of Sweden, other to equatorial Ecuador (included Pierre Bouguer) to comparelength of degree of arc at both sites.

• Pierre Simon, Marquis de Laplace: gravity obeys simple differential eq.

(early to mid 1800s) • Lord Cavendish, 1798, determined G, hence mass of Earth (estimate of G was

6.754x10-11)

Torque required to twist quartz fiber:

Torque provided by gravity:

Set equal and solve for G: (current value 6.6720x10-11 MKS) •


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• Cavendish experiment leads to mass, bulk density of Earth:

When mass causing acceleration, M, is Earth, we use g to representacceleration

We know R = 6371 km (how?), g = 9.8 m/s2, G = 6.67x10-11 MKS (whatare the units?), so M = 6.0x1024 kg(Bold numbers: memorize!)

Bulk density:

•

• • But how was the Newton defined? Improvement in accuracy of G (and

hence mass of Earth) over time:

http://geophysics.ou.edu/gravmag/history/cavendish_balance.jpghttp://geophysics.ou.edu/gravmag/history/newton.htmhttp://geophysics.ou.edu/gravmag/history/newton.htmhttp://geophysics.ou.edu/gravmag/history/cavendish_balance.jpg


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Gravity as Geophysical Tool

• Kater, 1818, reversible pendulum: absolute g • Earliest efforts to locate oil-bearing structures involved gravity: just before

1900, Baron Roland von Eötvös, Hungary o torsion, or Eötvös balance o measures distortion in g field from buried bodies o slow, cumbersome to operate

• 1915/16 torsion balance survey at 1-well oil field at Egbell, now inCzechoslovakia; highly successful

• 1917 Schweider: salt dome in Germany • 1922 Shell: Horgada field in Egypt • 1922 Spindletop field in Texas - salt structure • Vening Meinesz, 1928, shipborne pendulum • 1930s - Gulf Research & Development, 1st gravimeter (direct readings of

g differences; oil boom, LaCoste-Romberg, Worden meters patented

Potential Fields

Fields

A field is a set of functions of space and time.

We are concerned with 2 kinds of fields:

1. Material fields describe some property at a point of the material and at agiven time (intensive quantity)

Examples: density, porosity, magnetic susceptibility, temperature; not amaterial property: mass, heat; these are extensive quantities (depend onextent of material)

2. Force fields describe forces that act at each point of space at a giventime

Examples: gravity, magnetic field, electrostatic field

Fields can be scalar or vector or tensor

A vector field can be described in terms of field lines (or lines of flow, or lines offorce or flux lines). These are lines that are tangent at every point to the vectorfield.

Potential Theory


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Concept of potential

Example: Consider map of ski area: put arrows everywhere giving magnitudeand direction of slope; It is easier just to give elevation at each point!

In 1-D

In 3-D:

2-D example of relationship between scalar potential and vector field

[Note: ∇∇∇∇ is the "del" operator or gradient operator; it is always a vectorquantity; sometimes it is written with an arrow over it, or boldfaced, toindicate that it is a vector operator] Thus we see that a scalar field (elevation)can give rise to a vector field (slope)

Another example: temperature field (scalar), heat flow field (vector), where

Conservative Fields

For force fields (vector fields) it can be shown that if the force field isconservative, it may be (and must be) represented as the gradient of a scalarfield.

1. All force fields derived from scalar field are conservative 2. All conservative fields can be derived from scalar

http://geophysics.ou.edu/gravmag/potential/potential_vector_example.htmlhttp://geophysics.ou.edu/gravmag/potential/potential_vector_example.htmlhttp://geophysics.ou.edu/gravmag/potential/potential_vector_example.html


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Let's show that a force field derived from scalar is

conservative:

Conservative:

Stokes Theorem: [Kaplan, Advanced Calculus, 2ndEd., p. 344 ff.]

If there are no singularities in F, then U must be continuous and differentiable, soorder of differentiation doesn't matter, and

and therefore F is conservative.

Newton's Universal Law of Gravitation


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Newton realized that all of Kepler's laws regarding the motion of the planetscould be explained if a) the planets could be treated as point masses (and hewent off and invented integral calculus to prove this was a good approximation),and b) if the gravitational force between two objects was proportional to theproduct of their masses and inversely proportional to the square of the distance

between them:

In geophysics, we are interested in the force exerted on a "test body" by theEarth:

Since this force depends on the mass of the body, we can divide both sides bythe mass of the test body (equivalent to using a test mass of 1 unit mass):

For any point mass:


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Since any mass distribution can be broken intoinfinitesimal ''point masses" and since g is linear in m:

Is gravity conservative?

For point mass, m, observation point P, at a distance r from mass,


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Since 1/r has same value at beginning and end of loop, g is conservative. In

fact, by this same reasoning, all central force fields (like f(r), which onlydepends r) are conservative.

Potential as Work

Usually define potential as work required to bring unit mass from infinity todistance r from infinitesimal mass causing the potential:

Defined this way, potential is positive, and tends to zero as r goes to infinity (aswe get an infinite distance from the mass). [Note from Parasnis, 5th Ed., p. 60:"This is the same definition as that adopted by Kellogg (1953) in his classicbook on potential theory and (implicitly) by Jeffries (1976), among others.As defined in this manner, V has the units of [m2s-2] and represents thework done by the field per kilogram of a point mass m 0 when m0 movesfrom infinity to a distance r from m."]

Computing Gravity from Potential Field Finding g from U (Cartesian coordinates). With the definition of potentialgiven above, the acceleration of any point mass towards a mass, m, namelyGm/r2, is given by:


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Finding g from U (spherical coordinates):

Note on signs: defined this way, g will be negative, because it points in theopposite direction of the unit radial vector. For this reason, you sometimes see g defined as the positive gradient of potential, so that g (and |g|) will be a positivenumber, for convenience.

Integrating over masses to find total field

Because gravity is linear in mass (dm), we could find the gravitationalacceleration due to an extended body by vectorially adding (integrating) thegravity due to the infinite infinitesimal masses that make up that body, but thiswould be complicated. Because potential also depends linearly on mass (dm),and is scalar, integrating the potential over a body is easier. The potential due toseveral (even infinite) dm's is the sum (integral) of the potentials due to individualdm's. In Cartesian coordinates, for example,


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For an arbitrary mass distribution (Cartesiancoordinates)

For an arbitrary mass distribution (spherical coordinates)


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Example: what is potential due to sphere of density ρ?


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Deriving Poisson's and LaPlace's equations

First we will derive the Divergence Theorem and Gauss's Theorem

Divergence Theorem and Gauss's Theorem

Consider flux (flow) of material (force lines) through an infinitesimal box:


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The flux out of a volume V equals the divergence throughout volume V

Example: For incompressible fluid, flux is zero (no place for fluid to go), so

and since this is true for any arbitrary volume,


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Poisson's and Laplace's Equations

Pierre-Simon, marquis de Laplace, born

March 23, 1749, Beaumount-en-Auge,Normandy, France, died March 5, 1827,Paris. French mathematician, astronomer,

and physicist who is best known for his investigationsinto the stability of the solar system.

Spherical harmonics or Laplace's coefficients During the years 1784-1787 he produced somememoirs of exceptional power. Prominent amongthese is one read in 1784, and reprinted in the thirdvolume of the Méchanique céleste, in which he

completely determined the attraction of a spheroid ona particle outside it. This is memorable for theintroduction into analysis of spherical harmonics or Laplace's coefficients, as alsofor the development of the use of the potential - a name first given by Green in1828.

Siméon Denis Poisson, 1781 - 1840. Poisson's most important works were aseries of papers on definite integrals and his advances in Fourier series. Thiswork was the foundation of later work in this area by Dirichlet and Riemann.

In 1812 discovered that Laplace's equation is valid only outside of a solid.

For gravity,

Consider the net flux out of (or into) a closed volume:


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If M is inside the volume, the surface surrounding the mass takes up the entire"field of view", which is another way of saying that the total solid angle subtended

by the surrounding surface is 4π steradians (a steradian is the 3D equivalent ofa radian; the circumference of a unit circle is 2π, hence 2π radians in a circle.Similarly, the surface area of a unit sphere is 4π steradians).

More on solid angle...

(Laplace's Equation in Integral Form)

From Gauss's theorem,

http://geophysics.ou.edu/gravmag/laplace/solid_angle.htmhttp://geophysics.ou.edu/gravmag/laplace/solid_angle.htm


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Since this holds no matter how the volume is chosen,*

If M is outside the volume, total solid angle is 0 (2 ways to look at this: thesurface presents just as much of its front as its back, so they cancel, or noticethat the flux lines which go in one side of the volume bounded by the surfacecome out the other side, so the net flux is zero), so

Note that Laplace's equation is just the special case of Poisson's equation wheredensity is zero.

http://geophysics.ou.edu/gravmag/laplace/laplace.htmlhttp://geophysics.ou.edu/gravmag/laplace/laplace.html


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Applications of Poisson's Equation in Integral Form

1. Gravity due to spherically symmetric body: put imaginary surface("Gaussian surface") around the sphere

where M is the mass contained within the Gaussian surface.

2. Gravity inside a spherically symmetric hollow shell: put imaginarysurface ("Gaussian surface") anywhere within the hollow region around thesphere


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Since the mass contained within the Gaussian surface is zero,

Optional Assignment: Read Edgar Rice Burrough's At the Earth's Core

Homework problem: find gravity (g(r)) inside and outside a homogeneoussphere.

3. Gravity due to an infinite slab of thickness h and density ρρρρ: Bouguer'sFormula

• Consider "pill box" or cylindrical Gaussian surface• no flux out of sides of cylinder, by symmetry• g through top and bottom must be constant and perpendicular to top and

bottom (again, symmetry), so:

http://geophysics.ou.edu/gravmag/laplace/ECORE11H.HTMhttp://geophysics.ou.edu/gravmag/laplace/ECORE11H.HTM


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General Solution to LaPlace's Equation inSpherical Harmonics (Spherical HarmonicAnalysis)

• LaPlace's equation is , and in rectangular (cartesian)coordinates,

• In spherical coordinates, where r is distance from the origin of thecoordinate system, θ is the colatitude, and λ is azimuth orlongitude:

• Solutions to LaPlace's equation are calledharmonics • In spherical coordinates, the solutions would bespherical

harmonics

• Example: show that for point mass ( )


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Solving LaPlace's Equation

• Assume variables are separable: , so

• Multiply through by :

• Last term on LHS depends only onλ, yet first two do not dependon λ, so last term must be constant (and first two must add up tonegative of that constant).

This could be rearranged like this:

This is of the form where L(λ) has been replaced by x(t)and the constants "renamed." This is just theordinary differentialequation for the simple harmonic oscillator problem.

• This an ODE, with solution , where m isan integer

• Going back to the first two terms, we have

• Multiply through by :

• Again, terms must be independent, so both must be constant,giving this ordinary differential equation:

http://geophysics.ou.edu/gravmag/harmonic/harmonic.htmlhttp://geophysics.ou.edu/gravmag/harmonic/harmonic.html


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or

which has the solution

where l is any integer greater than or equal to 0

• Finally,

• This is another ordinary differential equation, known asLegendre's Equation, and has solutions of the form

, where are the Associated Legendre

Polynomials, are constants

Also, this is kind of neat: The Intel(R) Philanthropic Peer-to-Peer Program

• The general solution to LaPlace's Equation, then, is:•

http://www.intel.com/cure/http://www.intel.com/cure/http://www.intel.com/cure/http://www.intel.com/cure/


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• Examples of :

• Any Legendre polynomial can be found from thisgeneratingfunction:

•

• Spherical Harmonic Analysis consists of determining values for(and significance of) constants


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o

o for rotating Earth, might neglectλ dependence, i.e., allowonly m = 0 terms:

where are Legendre polynomials

• or, for convenience

1. Since a body that is finite in three dimensions (x, y, z) will "looklike" a point mass at infinity, the gravity must tend to GM/r2 as rgoes to infinity, so the potential will go to -GM/r. This eliminatesthe C'l

m, S'lm terms, because they depend on rl

2. For l = 0, m = 0, the legendre polynomial Plm(cos(θ)) (remember,

this is a function, not a constant times cos(r)) is 1, so C'00 is

identically equal to GM/r, where G is the Univ..., M is the mass ofthe body, and r is the distance from it. This term represents the"sphere" part of the potential.

3. If we set the origin at the center of mass of the body, there will be

as much mass east and west of the center of mass, north andsouth of the center of mass, and in front and behind the center ofmass. Therefore, the l = 1, m = 0 term must be zero, because it isasymmetrical between the northern and souther hemispheres.So, C1

0 = 0. This is because P00(cos(θ)) = cos(θ), which is positive

in the N and negative in the S (or vice versa, since C10, if it

weren't zero, could be negative).

•

• if we pick origin to be center of mass• , n odd, if equator is plane of symmetry, only true on

largest scale...


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• Other than the coefficients above which can be found from"common sense" boundary conditions, values for all the othercoefficients are determined by combined satellite and groundgravity data. The larger "longitudinally symmetrical" terms are:

o

o

o (oblateness)

o (pear-shapedness)

o

o

• measurements of Earth's gravity field show that the biggest effectis due to Earth's rotation and bulge

Why Do We Care About Spherical Harmonic Analysis of Earth's Gravity?

The most complete model for the earths gravitational field,based on an expansion in a Laplace series, is given by the GEM-T2 model. It contains 600 coefficients above degree 36:

More than you'd ever care to know about the GEM-T2 model...

The coefficients define the Earth's gravitational potential at any

point in space (outside the Earth).

• One way to visualize the potential field is to look at the shape ofan equipotential surface, usually (and conveniently) the onecorresponding to mean sea level. However...

• The l = 0, m = 0 term is the part of the Earth's potential that canbe explained by a perfectly spherically symmetric body (or pointmass). Think of it as a constant term.

• The l = 1, m = 0 term expresses the effect of oblateness (asmeasured by satellite, but not rotation, since satellites are not

affected by Earth's rotation, although surface gravitymeasurements clearly are). Although a thousandth as big as the l= 0, m = 0 term, it is about a thousand or more times bigger thanthe next biggest term.

• In order to even see the smaller terms on an equipotentialsurface, the oblateness of the Earth, known to have a flattening of1/298.25, must be subtracted from the "contour map."

http://geophysics.ou.edu/gravmag/harmonic/significance_of_harmonic_analysis.htmhttp://geophysics.ou.edu/gravmag/harmonic/significance_of_harmonic_analysis.htmhttp://geophysics.ou.edu/gravmag/harmonic/GEM-T2_model.pdfhttp://geophysics.ou.edu/gravmag/harmonic/significance_of_harmonic_analysis.htmhttp://geophysics.ou.edu/gravmag/harmonic/significance_of_harmonic_analysis.htm


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• Here is what the equipotential surface (geoid) looks like just using someof the lower degree and order terms:

International Gravity Formula

• accounts for variation of gravity with distance from equator• 2 effects:

o rotation of Earth (centripetal acceleration): ,

where


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o oblateness of Earth (caused by rotation)

This is the differential equation for the Simple Harmonic Oscillator

(SHO), or a mass on a spring:

where t is time, x is displacement of the spring, m is the mass and k isthe spring constant. The general solution to this equation is:

The undetermined coefficients A and B are determined by initialconditions (think of them as boundary conditions in time), namely theposition, x, and velocity v, of the mass, when t = 0.

Measuring GravityAbsolute Measurements

• Why?o ballistics, defenseo tectonic studies (e.g., glacial rebound)

http://geophysics.ou.edu/gravmag/measure/glacial_rebound.htmlhttp://geophysics.ou.edu/gravmag/measure/glacial_rebound.html


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o mass of Eartho tying relative measurements together

• more difficult to achieve high precision than relative• pendulum would work, but can't determine pendulum constant

accurately enough• therefore, use free-fall method

o photograph finely-etched meter stick illuminated by short-period, high-intensity flashes at precisely controlled timeintervals (~1 mgal)

o track time of fall of corner-cube reflector with laserinterferometer

o commercial instrument now available (see paper, Carter et al.,EOS)

o See also http://cires.colorado.edu/~bilham/GravFac.html o Iodine Stabilized Helium-Neon Laser

• Note: You don't have to assume an initial velocity and position ofzero. Given 3 positions and 3 times, you could solve for theacceleration (how?). In the actual experiments, they gathermany times and positions and then have an overdetermined system in

which they can both improve the accuracy of their acceleration

http://geophysics.ou.edu/gravmag/readings/absolute.htmlhttp://cires.colorado.edu/~bilham/GravFac.htmlhttp://geophysics.ou.edu/gravmag/measure/absolute.htmlhttp://geophysics.ou.edu/gravmag/measure/absolute.htmlhttp://cires.colorado.edu/~bilham/GravFac.htmlhttp://geophysics.ou.edu/gravmag/readings/absolute.html


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estimate but also estimate an error on that value.

• IGSN71 - network of absolute values• National Geodetic Survey, 1988, Leonard, Oklahoma:

Name Location Type g, microgals

Leonard AA Seismometer vault Primary absolute 979,720,911.7

Leonard CA Old non-magnetic building pier relative 979,720,997.0

Leonard NCMN NCMN relative 979,721,080.4

Leonard CB Leonard School relative 979,738,110.0

Measuring Gravity

Relative instruments

• pendulum measurements (still used atsea)


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for point mass on massless string,where I is moment of inertia about point of suspension; h is distance frompoint of suspension to center of mass. But, since K doesn't vary, can

measure change in g:o Used by Gulf R&D in 1930s; 1-second period (how long),

thermostat, vacuum o used in late 50s, early 60s by George Woolard at airports, seaports,

large cities, to establish worldwide gravity network

mass on spring

• static mass-spring system; k is spring constant:

for 0.1 mgal (10-6

m/s2

) accuracy:interferometer, wavelength of light about 5x10-7 m!

N.B. A 1oC change in T would change length of quartz spring 5.5microns! (How much of a change in gravity would this appear to be?)


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• consider system as SHO: or ,

but from above, , so

• thus, increasing the period increases sensitivity (but slows readings) • for ordinary mass-spring system, 20-s period requires 100 m spring/mass

system!

LaCoste & Romberg Zero-Length Spring

• History of the LaCoste & Romberg meter(s)

• • moment balance about pivot gives

• solving for g:

• we want to be small, and so a spring with anunstretched length of zero gives (theoretically) infinite sensitivity.

http://geophysics.ou.edu/gravmag/measure/lr_history.htmlhttp://geophysics.ou.edu/gravmag/measure/lr_history.html


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Worden Gravimeter


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Scintrex CG-3 Automated Gravity Meter

• "microgal" meter • manual levelling • automatic reading • automatic long-term drift correction


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• automatic tide correction • stores data

Underwater Gravimeters [From L&R web site]

• underwater meters operate on ocean bottom; no averaging as withshipboard meters

• usually shallow; can be modified for use at almost any depth (deep wateroperations slow, $$$)

• useful in swamps, on muskegs, frozen lakes, ice islands • if tranpsorted by helicopter, can hover over the gravity station while taking

reading • accuracy decreases with depth due to errors in measuring water depth

and position of meter • inherent precision meter about 0.01 mgal

• in actual sea operation, base station checks => about 0.1 mgal • water depth usually measured with pressure gauges; accuracy ~1/2% (0.6meter error in depth => ~0.1 mgal)

• overall accuracy of about 0.2 mgal is considered good in a survey in water160 meters deep


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Units

• Systeme International (SI; MKS): m/s2 • cgs: cm/s2 (gal) • milligal = mgal = 10-3 gal; microgal = µgal = 10-6 gal • gravity unit = gu = 0.1 mgal • 1 microgal = 0.7 mph/year!

•

• • typically desire survey accurate to 0.1 mgal (100 µgals); • g ~ 980,000,000 microgals! • to resolve anomalies, must adjust observations for several effects


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Our Meters


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History of the LaCoste & Romberg Meters

Dr. Arnold Romberg (1882-1974)[photo courtesy David Dillon, Jr.]

The LaCoste & Romberg Company was begunin 1939 by a graduate student at the Universityof Texas, Lucien LaCoste, and his facultyadvisor, Dr. Arnold Romberg. An interestingsidenote is that Dr. Romberg's grandson liveshere in Norman. He scanned this photo of hisgrandfather for me from a family scrapbook!

For a complete history of the development ofLaCoste & Romberg meters and their companysee:

http://www.lacosteromberg.com/meterhistory.ht

http://www.lacosteromberg.com/meterhistory.htmhttp://www.lacosteromberg.com/meterhistory.htm


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Airborne Gravity and Gravity Gradiometry

From the Grav-Mag mailing list:

Those of you who access the CSIRO web site should be wary of comparisons with the US Air Force

Gravity Gradiometer Survey System (GGSS), which we developed here in one of our previous incarnationsas Bell Aerospace. This technology is now more than fifteen years old and was an early exposure to using a

gravity gradiometer in an airborne environment. Since those initial flights in a C130 Hercules, much

experience has been gained operating different instruments in a variety of environments, which has resulted

in improved performance. In addition BHP and Bell Geospace have developed or improved upon software

techniques to optimize survey performance for their specific applications. Airborne results from the BHP

Falcon system give a better idea of current capabilities. Although we cannot make any independent

statements about BHP's Falcon gravity gradiometer performance, we believe that the specifications which

can provide a more accurate assessment of current performance can be found on the BHP web site:

http://www.bhp.com/default.asp?page=905

As described there, the Falcon system has been developed to operate in the higher turbulence experienced

at low terrain clearance (80m), thus improving gradient signal amplitude, which decays as the cube ofdistance from source. This substantially changes the CSIRO conclusions about the Eotvos sensitivity

needed to identify targets, since they anticipate 300m as a nominal altitude.

The Falcon instrument measures two curvature gradients. To create a full tensor, five gradient, airborne

instrument, the successful full tensor marine system is now being further developed to optimize it for

airborne applications. Lockheed Martin is currently inviting enquiries from organizations who would be

interested in participating in flight trials of such a system.

Andy Grierson

http://www.bhp.com/default.asp?page=905http://www.bhp.com/default.asp?page=905


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• can establish new base by looping from old base

Determining Elevations

• surveying (costs >= gravity survey!)

• topo sheet • inertial guidance system • altimeter • GPS, DGPS: GPS Primer • RTK dual-frequency DGPS (2 units with real-time

communication/correction between them) gives cm-level accuracy inseconds; cost (Feb. 2001) is about $45K

• DEMs 7.5-Minute DEM: 30x30-meter data spacing • More on DEMs (10 m vs. 30 m)

Determining Horizontal Position

• same as elevation • digitizer handy for getting latitudes

Adjusting Observed Gravity

Tidal correction

• secular variations in g are (generally) undesirable • effect of Sun about 50% of Moon • each has 12-hour period (front and back bulge), • but tidal inequality... • 2 effects:

o pull of bodies on meter o distortion of Earth (solid earth tide); adds about 12% to this effect

• total magnitude about 0.2, 0.3 mgal (refer to tidal correction lab) • to correct:

o fixed recording gravimeter located in or near survey area subtract variations from survey data probably most accurate correction, but $$

o tide tables (gravity) read tidal correction for given time and location many not apply well near water no longer published

o calculate tidal effect computer program yields correction for time and location incorporated into Scintrex meter

o include in drift correction

http://www.aero.org/publications/GPSPRIMER/index.htmlhttp://grid2.cr.usgs.gov/dem/dem.htmlhttp://geophysics.ou.edu/gravmag/reduce/10m_dem.jpghttp://geophysics.ou.edu/gravmag/reduce/tide-acd.txthttp://geophysics.ou.edu/gravmag/measure/scintrex1_smaller.jpghttp://geophysics.ou.edu/gravmag/measure/scintrex1_smaller.jpghttp://geophysics.ou.edu/gravmag/reduce/tide-acd.txthttp://geophysics.ou.edu/gravmag/reduce/10m_dem.jpghttp://grid2.cr.usgs.gov/dem/dem.htmlhttp://www.aero.org/publications/GPSPRIMER/index.html


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• data from Sandia's Lacoste and Romberg Model G meter, sitting inGilbert's lab, connected to chart recorder (light line); dark line from my tidecomputer program

Drift Correction

• secular variations in g are undesirable • instrument drift (DT, DP, creep), tides • assumptions

o changes are smooth and slow o changes are independent of location

• drift estimated by reoccupation of (base) station • must reoccupy every 2 hours or so:


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• using same base for entire survey not practical (or necessary):


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Station Time Dial Divisions Drift Rate Elapsed

Time Correction CorrectedReading

Base 11:20 762.71

GN1 11:42 774.16

GN2 12:14 759.72

GN3 12:37 768.95

GN4 12:59 771.02

Base 13:10 761.18

shift correction; the next day...

Station Time

Dial

Divisions Shift

Drift

Rate

Elapsed

Time Correction

Corrected

Reading

Base 10:20 763.68

GN5 10:42 775.16

GN6 11:14 765.42

GN7 11:37 765.35

GN8 11:59 770.32

Base 12:10 760.28

• Solution to above; don't look at this until you've tried it yourself! • 3-point drift correction

o assume ∆g = at2 + bt + c o evaluate a, b and c using ∆g known at one station at t1, t2 and t3o Excel spreadsheet example

Latitude correction

• Geodetic Reference System Formulae refer to theoretical estimates of theEarth's shape

• From these GRS formulae we obtains International Gravity Formulae(IGF)

• Several different formulae have been adopted over the years

http://geophysics.ou.edu/gravmag/reduce/quadratic_drift.xlshttp://geophysics.ou.edu/gravmag/reduce/quadratic_drift.xlshttp://geophysics.ou.edu/gravmag/reduce/drift.xls


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• In these equations, is geographic latitude and is commonly referredto as theoretical gravity or normal gravity

o First internationally accepted IGF was 1930:

o This was found to be in error by about 13 mgals; with advent ofsatellite technology, much improved values were obtained.

o The Geodetic Reference System1967 provided the 1967 IGF:

o Most recently IAG developed Geodetic Reference System 1980,leading to World Geodetic System 1984 (WGS84); in closed form itis:

• The IGF value is subtracted from observed (absolute) gravity data. Thiscorrects for the variation of gravity with latitude

International Gravity Formula "Calculator"

http://geophysics.ou.edu/gravmag/reduce/IGF_calculator.xlshttp://geophysics.ou.edu/gravmag/reduce/IGF_calculator.xls


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Latitude correction: short form

• note that over small range, curve is nearly straight-line slope

• where φφφφ is a typical latitude for the (small) field area • miles north, in above formula • subtract this amount for each mile north

Burger Table 6-1

Free-air correction

• "flag-pole" correction • accounts for decrease in g(r) due to change (increase) in elevation (r) • does not take in into account mass which may be elevating you

• could pick datum (e.g., sea level), compute "exact" difference, but oversmall changes in elevation, h, change is nearly linear

•

http://geophysics.ou.edu/enviro/grav/table6-1.xlshttp://geophysics.ou.edu/enviro/grav/table6-1.xls


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• note that, once IGF is removed, we take elevations relative to sea level,not center of Earth!

Taylor Series

• this is the free-air effect: g decreases (hence sign) with elevation • alternative "derivation":

• how big an effect? take g = 9.83 m/s2; r = 6371 km,

• correction: add 0.3086 times elevation in meters


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Bouguer correction

• "bulldozer" correction • accounts for mass (rock) responsible for elevation change (between

observation point and sea level, usually) • depends on density of material holding you up (Bouguer density) • gravity due to infinite slab, thickness h (m), density ρ (g/cm3):

• slab holding you up increases g; must subtract this effect out

• requires knowledge of h and ρ


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Burger Table 6-2

Selecting Reduction Density

• while sea level may be datum, variation in elevation occurs in near

subsurface • methods for selecting Bouguer density:

1. Standard Bouguer density = 2670 kg/m3

• nominally average crustal density • ensures continuity between surveys

2. Direct measurement

• collect samples, core, drill samples, hand samples

• inaccessibility; may not be representative

3. Geologic map to get rock type; get values from tables, graphs, etc.

• handbooks (physical properties of rocks and minerals) • sedimentary rock density histograms • rock density ranges • rock density means and ranges • salt density vs. sediment

Rock Type Density

Ice 880 - 920

sea water 1010 - 1050

Shale 1950 - 2700

limestone, dolomite 2500 - 2850

sandstone 2100 - 2600

soil & alluvium 1650 - 2200

rock salt 1850 - 2150

felsic igneous rocks 2550 - 2750

mafic igneous rocks 2700 - 3000 ultramafic rocks 3000 - 3300

4. Density profile (Nettleton method)

• collect closely-spaced g readings over topographic feature • make latitude, free-air correction

http://geophysics.ou.edu/enviro/grav/table6-2.xlshttp://geophysics.ou.edu/gravmag/references.htmlhttp://geophysics.ou.edu/gravmag/reduce/density1.gifhttp://geophysics.ou.edu/gravmag/reduce/density2.gifhttp://geophysics.ou.edu/gravmag/reduce/density3.gifhttp://geophysics.ou.edu/gravmag/reduce/salt.gifhttp://geophysics.ou.edu/gravmag/reduce/salt.gifhttp://geophysics.ou.edu/gravmag/reduce/density3.gifhttp://geophysics.ou.edu/gravmag/reduce/density2.gifhttp://geophysics.ou.edu/gravmag/reduce/density1.gifhttp://geophysics.ou.edu/gravmag/references.htmlhttp://geophysics.ou.edu/enviro/grav/table6-2.xls


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o must overcome small hole size, hostile environment, self-leveling

o "width" of investigation ("penetration into sides of borehole") ~ pointseparation

o best method for getting true formation density

6. Linear regression (least squares) method

• assumes no correlation between topography and subsurface density (i.e.,anomalies are randomly distributed with respect to elevation)

• therefore correlation between topography and g will be due to Bouguerslab

• plot ∆gfa vs. elevation, h • fit line through points

• slope will approximate 2πGρ; solve for ρ (Bouguer density)


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Least Squares Fit

• data pairs: x1, y1; x2, y2;...xn, yn, where n is the total number of data points • straight line: y = mx + b, where m is the slope, b the intercept


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• minimize the sum of the squares of the residuals (SSR):

• setting derivatives = 0 gives the normal equations:

• these two equations are used to solve for unknowns, m and b. Forhomework, show that:

• Problems with linear regression method for getting density: • if density varies (decreases in this example) with elevation, will get curve

• density may correlate with elevation; e.g., Arbuckles: limestones are hill-formers, shales are valley-formers...


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Bouguer correction at sea, underground

• surface survey

o first term replaces water with crust o second term Bouguer corrects to sea level

• underwater survey (for accurate value at sea)

• underground survey

Other corrections to gravity at sea

• not a stable platform as in land gravity

• accelerations can be of order , >> accuracy desired

• corrections also apply to airborne gravity


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Acceleration correction

• ship can't accelerate up or down very long, so average over t eliminates az (not so for aircraft)

• must know ax, ay (accelerometers, gyroscope to keep accelerometerslevel)

• natural period of land g meter typically ~10s, but waves also ~10s, so shipgravimeter must have much longer period


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Eotvos correction (Nettleton, p. 116 - 118)

• important for shipborne and airborne gravity • component of velocity in E direction increases apparent angular velocity

(Coriolis Force) • biggest single source of error in shipborne gravity comes from error in V, α

(although GPS, particularly DGPS, probably helps significantly) • centrifugal acceleration

• component in vertical direction

• effect in change in angular velocity, ω (∆ω will be small compared to ω onship or even plane)

• if velocity is V, E component is

• angular velocity due to this motion is


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• so, daV is given by

• there is also a simple acceleration in vertical direction due to total velocity,V

• Total Eotvos Correction

• For V in knots, correction in mgals

• Example: 10 knots E at equator => 75 mgals!

Terrain correction

• Bouguer correction assumes infinite slab; terrain correction corrects forthis erroneous assumption!

o

Bouguer correction works for gently sloping surfaces, liketopography on basement o error < 3% for slope < 1/5 (see Adams and Hinze, vol. 3, SEG Geotech. &

Environ. Gphy., p.99) • Terrain correction

o always positive o requires detailed info on elevation around station, not just at station


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o size of terrain correction depends on relief and its proximity tostation

Burger Table 6-3

Hammer Terrain Correction Chart

Terrain Correction with DEMs

• advent of DEMs has made medium to far-field correction much easier • short-field correction may be done using "newly available, reflectorless

laser rangefinders. Such rangefinders permit a detailed digital terrain datato be acquired in the vicinity of a gravity station within only 2-3 minutes,permitting the gravity meter operator to acquire the terrain data needed ...at the same time as the gravity measurements are made."

• Geodesy Group at Curtin University - Terrain Effect

http://geophysics.ou.edu/enviro/grav/table6-3.xlshttp://geophysics.ou.edu/gravmag/reduce/hammer.xlshttp://www.cage.curtin.edu.au/~geogrp/research/res-tc.htmlhttp://www.cage.curtin.edu.au/~geogrp/research/res-tc.htmlhttp://geophysics.ou.edu/gravmag/reduce/hammer.xlshttp://geophysics.ou.edu/enviro/grav/table6-3.xls


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• Geophysical Software • Sample DEM: Contours, Color Shaded, Shaded Relief, Map

Free-air, Bouguer, Isostatic Anomaly

• latitude and free-air correction virtually always made, giving the Free-AirAnomaly (FAA)

• for environmental/exploration work, some Bouguer correction is made

• for gravity in mgals and elevation in meters, then

• The Standard Bouguer Anomaly uses Bouguer density of 2.67 g/cm3

• note that, once ρ is chosen, FAC and BC can be combined into onecorrection: the elevation correction

• the Complete Bouguer Anomaly also includes the terrain correction

• sometimes make a correction for isostasy, giving Isostatic Anomaly

Estimating Survey Error

• dependent errors add algebraically • independent errors add vectorially • sources of error:

o meter dial o meter consistency o drift

o latitude o elevation (Free-air and Bouguer partially cancel) o terrain o others?

http://www.geopotential.com/http://geophysics.ou.edu/gravmag/reduce/norman_contour.gifhttp://geophysics.ou.edu/gravmag/reduce/norman_color.jpghttp://geophysics.ou.edu/gravmag/reduce/norman_dem.jpghttp://geophysics.ou.edu/gravmag/reduce/norman_quad.gifhttp://geophysics.ou.edu/gravmag/reduce/norman_quad.gifhttp://geophysics.ou.edu/gravmag/reduce/norman_dem.jpghttp://geophysics.ou.edu/gravmag/reduce/norman_color.jpghttp://geophysics.ou.edu/gravmag/reduce/norman_contour.gifhttp://www.geopotential.com/


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• determining error of calculated quantity

Error: A Calculus/Physics Refresher

Suppose you want to find the mass of a homogeneous planet of density ρ and

radius R. You know the volume of a sphere, and that the mass would be thevolume time the density, so you have:

Now, you know that you can never determine neither the density nor the radiusexactly, so you'd like to know how much error there would be the masscalculation if you make a (small) error in the density of radius.

Let's start with density. What we're really asking is, "how much of a change willthere be in mass if we have a small (technically, infinitessimal) change in density.

This should ring a bell from some math class you took. The quantityrepresents the (infinitessimal) change in mass due to a (infinitessimal) change indensity, with R constant. Performing the derivative, we get

This first equation will have the units of mass/density (which happens to bevolume). So if we know density to an accuracy of 100 kg/m3, and the radius ofthe planet is 6371 km,

and our error in mass turns out to be that amount. (For reference, the bulk

density of the Earth is about 5500 kg/m3, and the mass is about .

The error of 100 kg/m3 is about 1 part in 55, then, and so the mass is off byabout the same ratio.)

Now let's deal with an error in radius. Again, we want to know the expectedchange in mass for a small change in radius, so

http://geophysics.ou.edu/gravmag/error.htmlhttp://geophysics.ou.edu/gravmag/error.html


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Notice that the error in M due to an error in R now depends on R! In other words,if we make an error of 1 km for a big planet, the error in mass will be muchgreater than for a small planet. (Physically, you can picture the shell of additionalmaterial 1 km thick; its surface area will vary as R squared.)

Now, you might say, "if we don't know R precisely, how do we know what R touse in the formula?" And the answer is, you don't, but you can still estimate theerror, even though you won't know the amount of error exactly!


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Graphical Smoothing

• graphically estimate regional field• subtract regional from total to produce residual • time-consuming, very subjective (good and bad)

Profile

• sketch regional field• subtract regional from total to create residual

• profile, regional removal • another profile, regional removal

Contour Map

• sketch contours representing regional field• estimate regional values at control points• subtract regional values at control points from (total) values at control

points• re-contour residual control point

http://geophysics.ou.edu/enviro/grav/case_histories/regional_profile.gifhttp://geophysics.ou.edu/enviro/grav/case_histories/regional_profile2.gifhttp://geophysics.ou.edu/enviro/grav/case_histories/regional_profile2.gifhttp://geophysics.ou.edu/enviro/grav/case_histories/regional_profile.gif


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• contour data, smoothing • ring method, contour data • picking ring size

Polynomial Fitting, Trend Surfaces (method ofleast squares)

• fit a smooth (polynomial) surface to data to represent regional• subtract calculated regional value from observed value at each point to get

residual field• zeroth-order polynomial (constant, g0)

o minimize Sum of Squares of Residuals (SSR)

http://geophysics.ou.edu/enviro/grav/case_histories/regional_contour.gifhttp://geophysics.ou.edu/enviro/grav/case_histories/regional_ring.gifhttp://geophysics.ou.edu/enviro/grav/case_histories/regional_ring2.gifhttp://geophysics.ou.edu/enviro/grav/case_histories/regional_ring2.gifhttp://geophysics.ou.edu/enviro/grav/case_histories/regional_ring.gifhttp://geophysics.ou.edu/enviro/grav/case_histories/regional_contour.gif


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• first-order polynomial in x and y

o 3 equationso 3 unknowns (a, b, c)

• second-order polynomial in x and y

• north-central Iowa trend surfaces

Gridding irregularly spaced points

• most filtering and averaging schemes require regularly spaced (gridded)points

• can overlay grid on contour map, interpolate values at grid points (groan)• or, use computer program (like Surfer)

Inverse Distance Squared

• compute value at grid point based on neighbors• use nearest N points, or points within R of grid location• value at P is (inverse distance squared) weighted average of selected

neighbors

http://geophysics.ou.edu/enviro/grav/case_histories/least_squares.gifhttp://geophysics.ou.edu/enviro/grav/case_histories/least_squares.gif


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Kriging

• fit analytical surface to data points• use surface to compute values at grid points• "Kriging provides a means of interpolating values for points not physically

sampled using knowledge about the underlying spatial relationships in adata set to do so. Semivariograms provide this knowledge. Kriging isbased on regionalized variable theory and is superior to other means ofinterpolation because it provides an optimal interpolation estimate for agiven coordinate location, as well as a variance estimate for theinterpolation value."

Cubic Spline

• cubic spline is shape an elastic rod would take if constrained to fit atcontrol points

• bicubic spline is shape an elastic plate would take if constrained to fit atcontrol points

Minimum curvature

• find surface with least curvature that fits points withing a certain tolerance• requires odd number of grid points• Briggs, Ian C., 1974, Machine contouring using minimum curvature.

Geophysics, v. 39, pp. 39-47.• works well with profile data• works well with digitized contour data

Smoothing by averaging


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Running or moving average

• n-point running average

• here weighting factor is 1.0 for all points

Weighted averaging

• may want to weight closer points more (or less)

• in 2 dimensions (25-point average, for example):

• grid method of regional removal

• running average, profile • matrix smoothing (running average), contour data

Vertical Derivatives

• often take first or second vertical derivative of field

http://geophysics.ou.edu/enviro/grav/case_histories/regional_grid.gifhttp://geophysics.ou.edu/enviro/grav/case_histories/run_avg.gifhttp://geophysics.ou.edu/enviro/grav/case_histories/contour_smooth.gifhttp://geophysics.ou.edu/enviro/grav/case_histories/contour_smooth.gifhttp://geophysics.ou.edu/enviro/grav/case_histories/run_avg.gifhttp://geophysics.ou.edu/enviro/grav/case_histories/regional_grid.gif


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• enhances short wavelength anomalies relative to long wavelengthanomalies (?-pass filter)

• note, for modelling purposes, that result does not have units of g• tends to delineate edges of anomalous body

4-point average and the 2nd vertical derivative

• form residual by subtracting average of 4 closest points from center point


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• now consider Laplace's equation

• in discrete form, we use finite differences

• the first backward, first forward, and second central differences are

• similarly


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• so for the second vertical derivative (assume ∆x = ∆y)

• which differs from the 4-point grid average only by a constant• thus, second vertical derivative amounts to horizontal "curvature" of field• effect of 2nd derivative • radius of curvature and 2nd derivative • buried river channel [modelled gravity; Burger Fig. 6-31] • salt domes, Texas • LA Basin • Cement field, OK

Upward Continuation

• Stokes's Theorem: If gravity values are known everywhere onEarth's surface, gravity at any higher point can be calculated fromthese values

• knowing field at one elevation, can compute what field would look like at ahigher elevation (upward continuation) or lower elevation (downwardcontinuation

http://geophysics.ou.edu/enviro/grav/case_histories/regional_2nd_deriv_small.gifhttp://geophysics.ou.edu/enviro/grav/case_histories/regional_2nd_deriv2_small.gifhttp://geophysics.ou.edu/gravmag/filter/2nd_vert.gifhttp://geophysics.ou.edu/enviro/grav/case_histories/2nd_vert.gifhttp://geophysics.ou.edu/enviro/grav/case_histories/2nd_vert_la.gifhttp://geophysics.ou.edu/enviro/grav/case_histories/2nd_deriv_cement.gifhttp://geophysics.ou.edu/enviro/grav/case_histories/2nd_deriv_cement.gifhttp://geophysics.ou.edu/enviro/grav/case_histories/2nd_vert_la.gifhttp://geophysics.ou.edu/enviro/grav/case_histories/2nd_vert.gifhttp://geophysics.ou.edu/gravmag/filter/2nd_vert.gifhttp://geophysics.ou.edu/enviro/grav/case_histories/regional_2nd_deriv2_small.gifhttp://geophysics.ou.edu/enviro/grav/case_histories/regional_2nd_deriv_small.gif


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• this amounts to 1/r3 weighted averaging• downward continuation enhances near-surface bodies more than deeper

bodies, hence lessens effect of regional• consider effect on bodies at different depths, d1 and d2

original photo cropped

resized Lanczos filter resampling

Wavelength Filtering - the Fourier Transform

http://geophysics.ou.edu/gravmag/filter/photo1.jpghttp://geophysics.ou.edu/gravmag/filter/photo2.jpghttp://geophysics.ou.edu/gravmag/filter/photo3.jpghttp://geophysics.ou.edu/gravmag/filter/photo4.jpghttp://geophysics.ou.edu/gravmag/filter/photo4.jpghttp://geophysics.ou.edu/gravmag/filter/photo3.jpghttp://geophysics.ou.edu/gravmag/filter/photo2.jpghttp://geophysics.ou.edu/gravmag/filter/photo1.jpg


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High-Pass Filtering

Unfiltered: Queen: Bohemian Rhapsody

Passes high wavenumber components; attenuates low wavenumber or longwavelength components

Queen: Bohemian Rhapsody, high-pass at 2000 Hz

Low-Pass Filtering

Passes low wavenumber components; attenuates high wavenumber or shortwavelength components

Queen: Bohemian Rhapsody, low-pass at 1000 Hz

Bandpass Filtering

Passes a band of wavenumbers, i.e., frequencies between u1 and u2 are kept,

rest eliminated (or at least attenuated).

Queen: Bohemian Rhapsody, band-pass between 1000 Hz and 2000 Hz

http://geophysics.ou.edu/gravmag/filter/queen_bohem_rhap_hi2000.mp3http://geophysics.ou.edu/gravmag/filter/queen_bohem_rhap_lo.mp3http://geophysics.ou.edu/gravmag/filter/queen_bohem_rhap_band_1000_2000.mp3http://geophysics.ou.edu/gravmag/filter/queen_bohem_rhap_band_1000_2000.mp3http://geophysics.ou.edu/gravmag/filter/queen_bohem_rhap_lo.mp3http://geophysics.ou.edu/gravmag/filter/queen_bohem_rhap_hi2000.mp3http://geophysics.ou.edu/gravmag/filter/queen_bohem_rhap.mp3


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• consider, for simplicity, a 1-D function, g(x) (e.g., a gravity profile), in thespace domain, g(x), and wavenumber domain, G(u):

• high-pass filtering consists of passing (keeping) high wavenumber (shortwavelength) terms

• low-pass filtering consists of passing (keeping) high wavenumber (shortwavelength) terms


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• band-pass filtering consists of passing (keeping) only that portion of thewavenumber spectrum in the band u1


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• this mitigates Gibbs phenomenon - a "ringing" that results from too sharpa filter

• test• test low• test high• sara (key clicks)

• eastern Sierra Nevada, complete Bouguer, low-pass>50 km • eastern Sierra Nevada, high-pass>50 km • Garber oil field, Oklahoma • Regional field used • Residual Garber gravity

Other Filter Operations

Filter H(u,v)

Upward (z0) continuation,where z is continuation height (depth)

1st vertical derivative

2nd vertical derivative

[Note: The 2 π terms above may be absent in some texts, depending on whether

the Fourier transform has 1/(2 π ) term]

Directional Derivatives

• comments from Dr. Lyatsky on directional derivatives

Wavelength Filtering - Wavelet Processing

• Localized analysis of frequency content• used, e.g., for better compression of images (JPEG2, or Lurawave)

http://geophysics.ou.edu/enviro/grav/case_histories/reduce-c-d.gifhttp://geophysics.ou.edu/enviro/grav/case_histories/reduce-e-f.gifhttp://geophysics.ou.edu/gravmag/filter/garber2.gifhttp://geophysics.ou.edu/gravmag/filter/garber_regional_s.gifhttp://geophysics.ou.edu/gravmag/filter/garber2.gifhttp://geophysics.ou.edu/gravmag/filter/lyatsky.htmlhttp://geophysics.ou.edu/gravmag/filter/garber2.gifhttp://geophysics.ou.edu/gravmag/filter/garber_regional_s.gifhttp://geophysics.ou.edu/gravmag/filter/garber2.gifhttp://geophysics.ou.edu/enviro/grav/case_histories/reduce-e-f.gifhttp://geophysics.ou.edu/enviro/grav/case_histories/reduce-c-d.gif


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• notice that analysis of period is different for different time periods


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Interpretation of Gravity Data

Uniqueness (ambiguity) Problem

• applies to all potential field methods, and, indeed, all geophysical methods • there is an inherent ambiguity in interpretation of gravity data

• even if you had gravity at every point on Earth's surface, there are multiplemodels that would produce those values because of integral nature of gravity, itcan be proven that any anomaly can be result of an infinite number of densitydistributions!

• •


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Constraining Interpretations

• not hopeless: gravity eliminates even "more infinite" number of densitydistributions

• combine gravity data with other constraints: o density of crustal rocks, particularly in local area o configuration of rocks: well data, regional geology, etc. o other geophysics: magnetics, seismic, etc.

Interpretation approaches

Direct Interpretation: Inverse method

• only possible if many constraints (artificial?) imposed

1. assume general class of model (e.g., buried sphere) 2. analyze anomaly (anomalies) to define specific model

Indirect Interpretation: Forward modelling

1. assume specific initial subsurface density model 2. calculate gravity (always do-able, at least numerically) 3. compare with data 4. adjust density model as necessary 5. repeat steps 2 through 4

Role of interpretation in survey planning

• should the survey be conducted?! • how should it be conducted?

Simplifying density models

• because we are interested in (and indeed only measure) change in g, weare only interested in changes in density (density contrast)

• background density can always be subtracted

• furthermore, horizontal slab doesn't contribute to gravity anomalies • in some cases (e.g., sphere, horizontal cylinder), mass

excess/deficiency , is determinable quantity • a datum shift may be made to compare model to data


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• Another example illustrating the ideas of gravity anomaly and densitycontrast :

Vertical g

• gravity of Earth >> any anomaly • gravity defines vertical • therefore gravity meters only measure vertical component:


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Gravity due to simple bodies

• easiest, most versatile approach to survey planning, interpretation • even complex structures produce anomalies similar to simple shapes; for

example, horizontal circular cylinder versus square cylinder:


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Infinite Slab

, where d is the thickness

• works for gently sloping surfaces • example: topography on basement • error < 3% for slope < 1/5 (see Adams and Hinze, vol. 3, SEG Geotech. & Environ. Gphy., p.99 • use estimate magnitude of anomaly for many flat-layer situations:

o relief on density contrast boundary: basement, bedrock, etc. o dip-slip fault in horizontal strata o laterally extensive mines o water removal/recharge in horizontal aquifer


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Sphere

• applicable to approx. equidimensional bodies (longest dimension


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Where is g 1/2 of maximum?

"Inversion"

technique

1. find distance (x1/2) from peak of anomaly where anomaly is half maximumanomaly:

2. depth of body:

3. since


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Notes

• non-uniqueness:

(can't determine radius AND density contrast) • anomaly size is relative to "baseline", or "background" • in general, half-width to left and right are unequal

Example:

gmax = 35 mgals, 1/2-width at 17.5 mgals = 7.5 m; therefore z = 10 m; assumingdensity contrast of 1.0, find radius of sphere.


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Infinite Horizontal Cylinder

• applicable to bodies much longer in one horizontal direction than invertical or other horizontal direction


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• tunnels, river channels, horst or graben block, etc.


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• This is a max at x=0, or

where we've dropped the subscript v.

• Find the relationship between half-width and depth:


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Vertical Cylinder

• no simple expression exists for gravity off-axis; but on axis:

special case: infinite slab

special case: semi-infinite cylinder

Note that finite cylinder formula can be derived by superposition of two semi-infinite cylinders


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Narrow Vertical Cylinder

• while no simple expression exists for gravity off-axis of a thick cylinder, forthin cylinder an approximate solution exists

• good for z > 2a

semi-infinite case


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• depth criterion:

• knowing z, we can now find σ


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finite case

• use superposition:


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Semi-infinite Horizontal Slab

finite horizontal slab


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thin semi-infinite sheet

• for a line,

• so for a sheet,


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• Note that

• Depth criterion


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thin finite sheet

Computing depth:

• Compute at center of anomaly:

• Compute to one side:

• On graph with no vertical exaggeration, find depth which yields theseangles:


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2D Grids

The vertical gravity component due to a line element of mass σ per unit length is:

Now consider an arbitrary 2-D body of density ρ:


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The area of the shaded element is dz*dx, so


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For ∆θ, ∆z constant, each block contributes the same to g

"Computer" Methods of Interpretation

Talwani, 1973, in Bolt: Computational Methods in Geophysics


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3D laminar bodies

Talwani and Ewing, 1960, Geophysics, v. 25, 203-225 Pluoff, 1976, Geophysics, v. 41, 727-739

• uses solid angle approach • for thin horizontal sheet,

2-D polygon method Talwani, Sutton and Worzel, 1959, JGR, 64: 1545 - 1555Talwani, Worzel and Landisman, 1959, JGR, 64: 49-59

• uses line integral approach


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This is the method used in most 2D computer modelling programs, like GM-SYS.


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3D vertical prisms (method of Cordell and Henderson)

Three-dimensional iterative method of Cordell and Henderson(Cordell, L., and Henderson, R. G., "Iterative three-dimensionalsolution of gravity anomaly data using a digital computer,"Geophysics 33 (1968), 596-601). Block heights are relative to apredetermined reference surface; density contrast is alsopredetermined. Initial block height might be determined by usingthe gravity value above the block and using the Bouguer slab

approximation. Then slab heights are adjusted to give a best fit tothe measured gravity values (or a gridded gravity field derive frommeasured gravity data). Figure from Blakely, 1996.

• Danes, 1960, Geophysics, 25: 1215-1228 • square prisms, infinite bottom depth; get finite prisms by using another set

of prisms • iterative method:

o pick set of prisms o find g at center of each prism o adjust heights to match actual field o close to direct approach (inversion), given assumptions

General 3D Bodies

GRVMAG message from Manik Talwani re: his 3D G&M inversion program (5/10/2001)

http://geophysics.ou.edu/gravmag/references.htmlhttp://geophysics.ou.edu/gravmag/grav_interp/manik_talwani_3d.htmlhttp://geophysics.ou.edu/gravmag/grav_interp/manik_talwani_3d.htmlhttp://geophysics.ou.edu/gravmag/grav_interp/manik_talwani_3d.htmlhttp://geophysics.ou.edu/gravmag/grav_interp/manik_talwani_3d.htmlhttp://geophysics.ou.edu/gravmag/references.html


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• Bhattacharyya, B. K., Navolio, M. E., 1976, A Fast Fourier Transformmethod for rapid computation of gravity and magnetic anomalies due toarbitrary bodies: Geophys. Prosp., 24, 633-649.

• Gerard, A., Debeglia, N., 1975, Automatic three-dimensional modeling forthe interpretation of gravity or magnetic anomalies: Geophysics, 40 (6),

1014-1034. • Talwani, M., Ewing, M., 1960, Rapid computation of gravitational attractionof three-dimensional bodies of arbitrary shape: Geophysics, 25 (1), 203-225.

• Okabe, M. 1979, Analytic expressions for gravity anomalies due tohomogeneous polyhedral bodies and translations into magneticanomalies. Geophysics v44, p730-744.

Interpretation Examples

• Infinite Slab: Bedrock depths, Reading, Mass.

• Subsurface voids, Medford Caves, Florida and without vertical exaggeration

• Valley geometry, Pine Valley, central Nevada o location, Bouguer gravity map

http://geophysics.ou.edu/enviro/grav/case_histories/bedrock_depth2.gifhttp://geophysics.ou.edu/enviro/grav/case_histories/voids.gifhttp://geophysics.ou.edu/gravmag/grav_interp/fla_caves_no_vert_ex.gifhttp://geophysics.ou.edu/gravmag/grav_interp/fla_caves_no_vert_ex.gifhttp://geophysics.ou.edu/enviro/grav/case_histories/pine_valley.gifhttp://geophysics.ou.edu/enviro/grav/case_histories/pine_valley.gifhttp://geophysics.ou.edu/gravmag/grav_interp/fla_caves_no_vert_ex.gifhttp://geophysics.ou.edu/gravmag/grav_interp/fla_caves_no_vert_ex.gifhttp://geophysics.ou.edu/enviro/grav/case_histories/voids.gifhttp://geophysics.ou.edu/enviro/grav/case_histories/bedrock_depth2.gif


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o o gravity, geologic profiles

http://geophysics.ou.edu/enviro/grav/case_histories/pine_valley2.gifhttp://geophysics.ou.edu/enviro/grav/case_histories/pine_valley2.gif


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o o horizontal cylinder model

o

http://geophysics.ou.edu/enviro/grav/case_histories/pine_valley3.gifhttp://geophysics.ou.edu/enviro/grav/case_histories/pine_valley3.gif


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o double-cylinder model

o o 2-D polygon model

http://geophysics.ou.edu/enviro/grav/case_histories/pine_valley4.gifhttp://geophysics.ou.edu/enviro/grav/case_histories/pine_valley5.gifhttp://geophysics.ou.edu/enviro/grav/case_histories/pine_valley5.gifhttp://geophysics.ou.edu/enviro/grav/case_histories/pine_valley4.gif


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Documents

Potential Field Methods