SIO 226: SIO 226: Introduction to Introduction to

Marine GeophysicsMarine GeophysicsGravityGravity

LeRoy DormanLeRoy Dorman

Scripps Institution of OceanographyScripps Institution of Oceanography

Winter, 2013Winter, 2013

Goals:Goals:To leave you with some tools to help see To leave you with some tools to help see

the connection between observed the connection between observed gravitygravity

(1)density variations in the earth and(1)density variations in the earth and(2)The earth's shape(including (2)The earth's shape(including

topography), which is the basis for topography), which is the basis for estimation of bathymetry from satellite estimation of bathymetry from satellite

altimetry.altimetry.Much of the analysis of gravity is based Much of the analysis of gravity is based

on a phenomenon called “upward on a phenomenon called “upward continuation”.continuation”.

We start with the terminology, the types We start with the terminology, the types of gravity anomalies, and what they of gravity anomalies, and what they

show us.show us.

Gravity ReductionsGravity ReductionsThe first task is to remove the earth's main field, The first task is to remove the earth's main field, which is much larger than the anomalies we which is much larger than the anomalies we

study. The formula of 1967 isstudy. The formula of 1967 is

Where phi is latitude. This is based on a rotating Where phi is latitude. This is based on a rotating ellipsoidal earth.ellipsoidal earth.

To correct for altitude difference from the ellipsoid, To correct for altitude difference from the ellipsoid, we use the derivative, -0.3086 mGal/meter. The we use the derivative, -0.3086 mGal/meter. The result of these two corrections is the Free Air result of these two corrections is the Free Air

anomaly.anomaly.To correct for the material between the point and To correct for the material between the point and

observation, we make the Bouguer plate observation, we make the Bouguer plate correction of 2correction of 2πρπρGhGh, which is 0.1119h mGal/m, , which is 0.1119h mGal/m, for a density of 2670kg/m^3. The result of for a density of 2670kg/m^3. The result of making this additional correction is the Bouguer making this additional correction is the Bouguer anomaly. anomaly. ρρ can be adjusted to optimize can be adjusted to optimize topographic suppression, thus estimating topographic suppression, thus estimating ρρ..

g ϕ=978031.846 (1+ 0.005278895 sin 2ϕ+ 0.000023462 sin 4ϕ)mGal

Eliminate Things We Eliminate Things We KnowKnow

Elevation: Free Elevation: Free Air reduction Air reduction add 0.3088 add 0.3088 mGal/mmGal/m

Topography: Topography: Bouguer and Bouguer and terrain terrain reductions reductions subtract 0.1967 subtract 0.1967 mGal/mmGal/m

Net: +0.1121 Net: +0.1121 mGal/mmGal/m

Isostasy: (later)Isostasy: (later)

Continent-scale DataContinent-scale Data Things which worked on a local Things which worked on a local

scale leave us with large anomalies scale leave us with large anomalies when applied to a larger data set.when applied to a larger data set.

The horizontal scale of the previous figure was ~1 km.The one below is ~1000 km and provides a very different picture.

Mantle Bouguer AnomalyMantle Bouguer Anomaly

At sea, the plate correction (evaluated for field at At sea, the plate correction (evaluated for field at the sea surface for topographic height directly the sea surface for topographic height directly beneath the ship) is a poor approximation to the beneath the ship) is a poor approximation to the field from the actual topography, so real field from the actual topography, so real bathymetric data are used.bathymetric data are used.

To reveal features of the mantle, Kuo and Forsyth To reveal features of the mantle, Kuo and Forsyth (1988) suggested removal of the gravity field from (1988) suggested removal of the gravity field from the crust, using whatever data are available- from the crust, using whatever data are available- from a seismic survey or assuming a simple constant-a seismic survey or assuming a simple constant-thickness crust. The usefulness of this, of course thickness crust. The usefulness of this, of course depends on the accuracy of the estimate of the depends on the accuracy of the estimate of the crustal field, which, in turn, depends on the crustal field, which, in turn, depends on the survey data used and other assumptions.survey data used and other assumptions.

Basics from Physics IBasics from Physics I

Gravity is a vector force per unit mass caused by the presence of other masses. For two masses, this force is, from Newton's law,

F=Gm1m2

l2.

The theory which describes gravity is called potential theory, since this force is the gradient (derivative) of a scalar potential,

and scalars are simpler to deal with. Since taking a derivative of both sides of an equation leaves us with a valid equation, many transformations can be used on the potential apply as well tocomponents of gravity.

U=Gml

Basics from Physics IIBasics from Physics IIGauss's Theorem is that Gauss's Theorem is that

which implies that which implies that

This means that we can choose a This means that we can choose a which produces any value of which produces any value of we desire. Thus we have no we desire. Thus we have no

hope of extracting the earth's hope of extracting the earth's density structure uniquely from density structure uniquely from

gravity data alone.gravity data alone.

∫∫ F⋅ds=−4 πG MF perp ( x)=−2 πG σ ( x)

σ(x)F perp ( x)

Laplace's Laplace's EquationEquation

In mass-free space, Laplace's In mass-free space, Laplace's equation applies, and this equation applies, and this

provides a strong constraint on the provides a strong constraint on the potential field potential field UU. Writing this out, . Writing this out,

we getwe get

and after Fourier transformation,and after Fourier transformation,

or,or,

∇ 2U=0

∂x1

2 U + ∂ x2

2 U + ∂ x3

2 U=0

−k 12U (k )−k 2

2U (k )−k 32U (k )=0

k 12+ k 2

2+ k 32=0

Fourier Fourier Transform and Transform and

derivativesderivativesIn one dimension, the Fourier In one dimension, the Fourier Transform of a function isTransform of a function is

The FT of a deriv isThe FT of a deriv isIntegration by parts,Integration by parts,with and with and so ,andso ,and

so multiplication by corresponds to so multiplication by corresponds to differentiation, since we can omit the differentiation, since we can omit the 2nd term.2nd term.

F x [ f ' (x)](k )=∫−∞

∞

f ' (x)e−2π i k x dx

∫ v du=[uv ]−∫ u dv

F (k )=∫−∞

∞

f (x)e−2π i k x dx.

du= f ' (x)dx v=e−2π i k x

u= f (x) dv=2π i k e−2π i k x dxF x [ f ' (x)](k )=[ f (x)e−2π k x ]−∫

−∞

∞

f (x)(−2π i k e−2π i kx dx )

2π i k

Fourier Transformation Fourier Transformation in 2Din 2D

The Fourier The Fourier Transformation is Transformation is where the real part of the where the real part of the kernel is plotted at the kernel is plotted at the right for one value of right for one value of kk. . The direction of The direction of k k is is normal to the wave normal to the wave “crests” and the “crests” and the magnitude of magnitude of kk is is 22π̸λ, π̸λ, where where λλ is the wavelength. is the wavelength.

F (k )=∫∫ f ( x)e i k⋅x dx1 dx 2

Upward ContinuationUpward ContinuationIn geophysical surveys, we normally In geophysical surveys, we normally

gather data in two dimensions, gather data in two dimensions, and , which transform into and and , which transform into and . The equation . The equation

however, tells us that if we however, tells us that if we know any two, we can calculate the know any two, we can calculate the

third!third!Thus if we have the gravity field at Thus if we have the gravity field at

some level in the earth (such as some level in the earth (such as from a lamina), we can calculate from a lamina), we can calculate

the field at the surface.the field at the surface.

k 12+ k 2

2+ k 32=0,

x1k 2k1

x1

Upward decay of anomaly Upward decay of anomaly amplitudeamplitudeIf we take a 2D Fourier transform of If we take a 2D Fourier transform of

a spatial grid of data, we end up a spatial grid of data, we end up with a 2D array of amplitudes with a 2D array of amplitudes

evaluated at two wavenumbers evaluated at two wavenumbers

When we reconstruct the gravity When we reconstruct the gravity field, the enter the calculation field, the enter the calculation

as the exponential kernel of the as the exponential kernel of the Fourier transform. A real value of Fourier transform. A real value of k k produces a sinusoidal variation but produces a sinusoidal variation but

an imaginary an imaginary k k produces an produces an exponential decay. So long exponential decay. So long

wavelengths decay slowly and short wavelengths decay slowly and short wavelengths decay rapidly.wavelengths decay rapidly.

k 32=−(k 1

2+ k 22)where k i=2π / λ i

k i

Simple modelingSimple modelingWe can calculate the gravity field of a We can calculate the gravity field of a geological model in 4 steps.geological model in 4 steps.

(1) Approximate the structure using one (1) Approximate the structure using one or more lamina using the equivalent or more lamina using the equivalent

stratum.stratum. (2) Make a Fourier expansion of the (2) Make a Fourier expansion of the

field from the model.field from the model.(3) “Upward continue” the Fourier (3) “Upward continue” the Fourier

components.components.(4) Synthesize the anomaly from the (4) Synthesize the anomaly from the

Fourier components.Fourier components.

Seamount Example

Seamount radius:10km

Water depth:4km

Ridge Example

Water Depth:4km

Is this a realisticdepth?

A Simple Isostatic A Simple Isostatic ModelModel

Consider a simple model Consider a simple model in which the topography in which the topography is mirrored by a is mirrored by a structure of like density, structure of like density, but opposite sign, at but opposite sign, at some (initially one) some (initially one) depth.depth.

This model is consistent This model is consistent with the the data, both with the the data, both local and continental, local and continental, which we have just seen.which we have just seen.

Implications of this Implications of this modelmodel

We represent topography as a lamina with density We represent topography as a lamina with density σ(σ(x)=x)=ρρh(x) and the Compensation is a lamina with h(x) and the Compensation is a lamina with density -density -σσ(x).(x).

Take one harmonic component at a time, Take one harmonic component at a time, a sinusoid with wavelength k =2π/λ, whose a sinusoid with wavelength k =2π/λ, whose gravity field is gravity field is

The field from the compensating lamina is The field from the compensating lamina is and the BA is and the BA is

Dividing by produces Dividing by produces which plots on a log scale as a line whose slope is which plots on a log scale as a line whose slope is

h (x)=h ei k 1⋅x1

g B(x)=2πρGheik 1 x1

gQ(k )=−2πρGheik1 x1e−k 1 zc

g BA(k )=−2πρGheik 1 x1e−k 1 zc

g BA g B Q(k1)=−e−k 1 zc

−zc

Isostatic Response DataIsostatic Response Data

Surveyor data is Surveyor data is oceanic from off oceanic from off Oregon, McNutt, 1978.Oregon, McNutt, 1978.

USA is from Lewis and USA is from Lewis and Dorman, 1970.Dorman, 1970.

Australia is Mcnutt, Australia is Mcnutt, 1978.1978.

Plotted on a log scalePlotted on a log scale

Shows straight Shows straight lines, more or less.lines, more or less.

Slope of oceanic is Slope of oceanic is shallower, since shallower, since compensation compensation depth is smaller.depth is smaller.

Circles are Circles are oceanic, squares oceanic, squares and triangles are and triangles are US and Australia.US and Australia.

-1

0

Idealized ModelsIdealized Models In terms of out simple model, the classical In terms of out simple model, the classical

Pratt and Airy models look like:Pratt and Airy models look like:

Mass Anomalies and Mass Anomalies and GeodesyGeodesy

A mass anomaly A mass anomaly not only changes not only changes the gravity, it the gravity, it changes the shape changes the shape of the earth's of the earth's geoid as well.geoid as well.

Fortunately, the Fortunately, the changes in the changes in the geoid are for geoid are for λλ > > 400 km.400 km.

In The Space DomainIn The Space Domain Convolving the gravity Convolving the gravity

anomaly field with the anomaly field with the Stokes functions Stokes functions produces the geoid produces the geoid height.height.

Since these functions are Since these functions are not localized around the not localized around the computation point, computation point, gravity at distant parts gravity at distant parts of the earth is important.of the earth is important.

Principle of SIO's Bell Principle of SIO's Bell GravimeterGravimeter

which is a force-which is a force-balance balance accelerometer.accelerometer.

Its accuracy Its accuracy depends on a depends on a very accurate very accurate digital digital voltmeter.voltmeter.

Its principle of Its principle of operation is operation is shown at right.shown at right.

Bell Gravimeter (reality)Bell Gravimeter (reality) In operational In operational

reality, its reality, its appearance is a appearance is a little different.little different.

Summary:Summary: Gravity measurements allow us to:Gravity measurements allow us to: measure the density of the surficial measure the density of the surficial

material, material, measure the shape of the earthmeasure the shape of the earth Show that mountains “float”Show that mountains “float”