The Earth as a PlanetIt is important to look at the Earth in its context as only one of the nine planets which make up the solar system.

This is an introduction to the Solar System, its formation, and composition, with special emphasis on the "terrestrial" or "inner" planets. I take a kind of historical approach, noting the patterns and regularities observed, for example, by Tycho Brahe, described by Johannes Kepler, and explained by Sir Isaac Newton. Laplace and even the philosopher Immanuel Kant figure into shaping our modern-day notions of the origin and composition of the solar system.

"Early attempts to explain the origin of this system include the nebular hypothesis of the German philosopher Immanuel Kant and the French astronomer and mathematician Pierre Simon de Laplace, according to which a cloud of gas broke into rings that condensed to form planets." - Encarta, http://encarta.msn.com/encyclopedia_761557663/Solar_System.html

Let's start by looking at some patterns in the solar system:

Titius-Bode LawTaking the point of view of a first-time visitor, one of the first things you would notice about the Solar System is that the spacing between the planets' orbits consistently increases as you move away from the Sun (with one exception). Furthermore, it's not a linear increase, so we need essentially two figures, at different scales, to represent the solar system (pictures courtesy of "The Nine Planets"):

Titius-Bode's law: Distance, r, of the nth planet from the Sun (in A.U.s) is given by:

rn = 0.4 + 0.3 x 2n

Planet n rn Actual

Mercury -infinity, -1 0.4, 0.55 0.39

Venus 0 0.7 0.72

Earth 1 1.0 1.0

Mars 2 1.6 1.52

Asteroids 3 2.8 -

Jupiter 4 5.2 5.2

Saturn 5 10.0 9.6

Uranus 6 19.6 19.2

Neptune 7 38.8 30.1

Pluto 8 77.2 39.4

The Titius-Bode law was used to help discover Ceres, a 1000 km asteroid, in 1802, and Uranus in 1781.

Check satellites of Saturn and Jupiter Law has never been given a scientific foundation, and may be "chance,"

combined with the fact that the outer planets are larger (more on why later), and hence "take up more room."

May have hindered models of Solar System formation as an arbitrary constraint (Brahic, Formation of Planetary Systems)

Orbital, Rotational Consistencies1. All of the planets have nearly circular orbits around the sun.2. All of the planets orbit the sun in a counterclockwise direction when

viewed from the "North."3. All but two of the planets spin on their own axes in a counterclockwise

direction.4. The sun spins on its own axis in a counterclockwise direction.

Nebular theory for the origin of the solar system (Laplace, Kant)

Solar system begins as a nebular cloud with some slow rotation. It consists of about 98% H, about 2% He, and small amounts of heavier elements

Gravity causes the cloud to collapse or condense Conservation of angular momentum causes the cloud to spin faster and

faster. In the plane of rotation, centrifugal force partly counteracts gravity;

perpendicular to this plane, gravity forces the cloud to take on the shape of a disk. (spinning pizza dough)

Approximately one half of the nebular cloud’s mass collapses to form the Sun.

Due to tremendous heat and pressure, fusion kicks off in the Sun. The Sun goes through a T. Tauri stage, blowing most of the rest of the

nebular cloud’s mass out of the solar system. Within the remaining nebular cloud are local concentrations of mass and

rotation (eddies); these condense to form the planets.

An object bigger than Pluto has been found in the outer solar system by Mike Brown (Caltech), Chad Trujillo (Gemini Observatory), and David Rabinowitz (Yale University). It is possible, perhaps likely, that it will eventually be considered to be our solar system's tenth planet. For more info, see NASA's press release and the discoverer's web site. Its temporary designation is 2003UB313; an official name will be given in due course (more). Brown et al have now also spotted a moon orbiting this object.

homework neatness

About "never drilling into the mantle: IODP Drillhole

Inner (Terrestrial) planets vs. outer (Jovian) planets

High temperatures near the Sun allowed only the most refractory elements could condense. (Refractory literally means “capable of enduring high temperature,” i.e., elements with a high condensation and melting point.)

Includes Fe and silicate minerals. Farther from Sun, lighter elements could condense, e.g., water (ice),

ammonia ice, etc. Larger outer planets had gravity to retain H, He (but also Fe, silicates) Terrestrial planets:

Smaller Denser Primarily Fe (and Ni) and silicates

Jovian planets: Larger Less dense Primarily H, He, with icy moons

A sidereal day is 23 hours 56 minutes and 4.09 seconds long.

si·de·re·al [sahy-deer-ee-uhl] adjective 1. determined by or from the stars: sidereal time. 2. of or pertaining to the stars.

"Jupiter is so big that all the other planets in our Solar System could fit inside Jupiter (if it were hollow). "

Tycho Brahe (1546-1601)

Danish astronomer

Compiled data on planetary orbits

Used astrolabe (no telescopes!)

[Galileo (1564 - 1642) introduced telescope to astronomy in 1609]

Kepler’s Laws (1571-1630)Danish astronomer, Brahe’s assistant

Developed Kepler’s 3 laws:

LAW 1: The orbit of a planet/comet about the Sun is an ellipse with the Sun's center of mass at one focus

This is the equation for an ellipse:

LAW 2: A line joining a planet/comet and the Sun sweeps out equal areas in equal intervals of time

LAW 3: The squares of the periods of the planets are proportional to the cubes of their semimajor axes:

Sir Isaac Newton (1643-1727)From Kepler’s laws (not an apple) Newton conceived of Universal Law of Gravitation:

For example, consider planet in circular orbit around Sun:

(Equation 1) - to be used in homework to find period of orbit as a function of orbit radius, etc.

Mass of the EarthFrom Newton's Universal Law of Gravitation, it would seem apparent that we could find the mass of the Earth. Take M to be Earth's mass, and m to be a "test mass." Since F = ma, we can divide both sides of the equation below by m to get the acceleration of gravity, ag:

Assuming our test mass is at the surface of the Earth, r is the radius of the Earth. In 200 B.C., Eratosthenes used shadows of a vertical stick at two different latitudes to determine that the radius of the Earth was 250,000 stadia. Unfortunately, we do not know which one of the various measurements used in antiquity is represented by the stadia of Eratosthenes. According to the researches of Lepsius, however, the stadium in question represented 180 meters, giving a radius of the Earth of 7,160 km. Today we know the mean radius of the Earth is roughly 6,371 km. Measuring the acceleration of an object just requires accurate measure of time (Galileo used water clocks). [Galileo and the Leaning Tower of Pisa]  So, we can find GM, but not M independently!

The Cavendish Experiment - "Measuring the Mass of the Earth" - 1798Henry Cavendish (1731-1810) developed an apparatus for experimentally determining the value of G involved a light, rigid rod which was 6-feet long. Two small lead spheres were attached to the ends of the rod and the rod was suspended by a wire. The angle of rotation yields the amount of torsional force. A diagram of the apparatus is shown below:

[How were the masses of the spheres determined?] It was not until Cavendish determined that G had a value of 6.75 x 10-11 N m2/kg2 that the mass of the Earth was known! Today, the currently accepted value of G is 6.67259 x 10-11 N m2/kg2. And this results, as you will show in homework, in:

Commercial Cavendish Apparatus "Big G" vs. "time"

This is perhaps the most important constraint on the composition of the Earth.  Any model for the composition and structure of the Earth must result in this mass. Is the Earth made of green cheese? Well, does (density of green cheese) x (volume of Earth) = M??

Bulk Density of the EarthOnce the mass of the Earth was established, its bulk, or average, density could be determined:

We know that the average density of continental crust is about 2670 kg/m3 and the average density of oceanic crust is about 3000 kg/m3 . Furthermore, from meteorites, among other things, we know the density of mantle material (peridotite) is about 3300 kg/m3. This would seem to suggest that the deeper interior of the Earth must have a very great density if the average is to be 5540 kg/m3. However, this density cannot be used to compare with the previously mentioned densities because they are measured at STP. The Earth's bulk density is "inflated" because of the tremendous pressures in the interior of the Earth. Although we will discuss later how it is done, the uncompressed bulk density of the Earth is about 4000 kg/m3. Still, the argument holds that the interior of the Earth must have a density significantly higher than mantle density.

Planet DensitiesBy observing the orbital radius and period of a natural or artificial satellite about a planet, its mass can be determined. Its radius is determined by astronomical observation. Once the mass and radius of a planet are determined, bulk density can be estimated:

Planet (kg/m3) u (kg/m3)

Mercury 5420 5300

Venus 5250 3900

Earth 5520 4000

(Moon) 3340 3340

Mars 3940 3700

(Asteroids) 3710 3710

Jupiter 1310 N.A.

Saturn 690 N.A.

Uranus 1190 N.A.

Neptune 1660 N.A.

Pluto 2080(?) ?

Jovian and Saturnian Satellites

Jupiter has 4 major (Galilean) satellites; this multi-satellite system has been referred to as a "mini solar system"

Galilean Moon103 km from

Jupiter Moon radius

Io 3570 421 1,815

Europa 2967 671 1,569

Ganymede 1940 1,070 2,631

Callisto 1865 1,883 2,400

Calculated and Actual Temperature of Planetary Surfaces

affected initial compositions affects current tectonics (cf. Earth, Venus, Mars) affects current "climatic" conditions, ergo existence of water, life, etc.

depends on distance from Sun and assumptions about planet's (moon's) surface reflectivity (bond albedo)

Energy received by a planet is proportional to cross-sectional area of planet, πR 2 :

As a result of Ercv, planet heats up and re-radiates energy; reaches steady state (heat received equals heat radiated, Erad).   Now, let's assume planets re-radiate that energy over their entire surface (because they are rotating, entire surface "shares" this energy, and that planets can be assumed to be perfect black-body-radiators.

Black-body Radiator: idealized surface for which relationship between T and radiated energy may be derived from thermodynamic (statistical mechanics) first principles:

The total surface of a planet is so total radiated energy is:

Setting received and radiated energies equal:

Example: Earth - r = 149.6 x 106 km = 1.496 x 1011 m  so  T = 278o K = 5o C  (very close)

Moon, Mercury, Mars, Earth are close to "predicted" Venus is much hotter, due to ? outer planets (Jovian planets) only roughly approximate black bodies

Actual Temperatures (oK)

CalculatedPlanet Day Night Mean

Mercury 700 100 452 444

Venus 721-731 732 730 323

Earth 277-310 260-283 281 278

Moon 380 100 280 278

Mars 240 190 215 223

Jupiter 120-150 - 120 121

Saturn 120-160 - 88 90

Uranus 50-110 - 59 63

Neptune 50-110 - 48 50

Pluto - - 37 44

For more information, see this site and this one

"Linear" vs. Rotational Mechanics

Moment of Inertia determined from deviations of satellite's orbit from conical section, or

precession of equinoxes mass is 0th moment of density:

center of mass found from 1st moment of inertia is second moment of density

Moment of inertia for some ideal bodies

Ideal Body (in order of increasing central concentration) I, moment of inertia

planet, mass m, in orbit of radius r 1.0 mr2

ring, radius R, mass m, spinning about sym. axis 1.0 mR2

hollow sphere, radius R 2/3 mR2

homogeneous sphere, radius R 2/5 mR2

sphere, core radius 1/2R, core density = 2 x mantle density 0.367 mR2

mass concentrated on axis 0.0 mR2

Precession of the Equinoxes

Animation of Precession of Equinoxes

change in position of Earth's spin axis ("seasons") w.r.t "fixed stars" caused by torque of Sun on Earth's equatorial bulge allows estimation of C-A (C: I about spin axis;  A: I perpendicular to that) period: 26,000 years thus, in 13,000 years, winter will be in July, if calendar is fixed to

position of Earth and Sun w.r.t. fixed star reference frame one part of Milankovitch Cycles

this (combined with spherical harmonic analysis of Earth's gravity) is how we determine moment of inertia of Earth, kMa2, where k is the gyrational constant (0.33078)

measure of central condensation of mass (second moment of mass) important constraint on radial distribution of mass in Earth for Earth

models (e.g., A&Z's PREM)

 "Real" Planets

Body I/mr2

Moon 0.391

Mars 0.365

Earth 0.3307

Neptune 0.29

Jupiter 0.26

Uranus 0.23

Saturn 0.20

Sun 0.06

Moon may have homogeneous distribution (note non-uniqueness), small Fe core at most

Mars has some central concentration, Fe core Earth's I consistent with large Fe core (I very well known; very important

constraint on density models, as is M!) Gaseous giants, and Sun, have r "inflated" by light gasses, thus

reducing I/mr2 value for Mercury?

Milankovitch Cycles precession of equinoxes (26,000 year period) change in inclination of the ecliptic (41,000 year period) change in orbital eccentricity (96,000 year period) observed in sedimentary record may be responsible for timing of Ice Age (Pleistocene) glaciations

Copyright J. L. Ahern 2009

Gravity and the Figure of the EarthThe Size of the Earth

see "Earth as a Planet" notes and section 2.11 of Lowrie

The Figure of the Earth (geodesy)

Jean Richer, 1672: pendulum clock, accurate in Paris, lost a 2 1/2 minutes per day in Cayenne, French Guiana

Newton, 15 years later (1687, Philosophiae Naturalis Principia Mathematica) correctly interpreted as due to oblate Earth (pumpkin) caused by centrifugal force due to Earth's rotation

Newton, assuming homogeneous Earth: flattening = (c-a)/a = 1:230 (~0.5%); actual 1:298 (~0.3%)

implies length of degree of meridian arc should subtend a longer distance in polar regions than near equator

French, based on traverse in France, believed earth was prolate (rugby ball)!

French Academy of Sciences, 1736-1737 sent expedition to measure degree of meridian arc in Lapland (Sweden), another, 1735-1743, measured 3 degrees of arc in Peru.

Concluded Earth indeed oblate a = 6378.136 km c = 6356.751 km R = 6371.000 km (a-c)/a = 1/298.257 a-R = 7.1 km R-c = 14.2 km

Figure of the Earth and its Rheology

A fluid with the Earth's density distribution, rotating once every day, would have a flattening of almost exactly observed 1:298.25 flattening.

Is Earth fluid? Or was it fluid?

Gravity Gravity is a vector quantity, with direction and magnitude Often treated as scalar because we generally can only measure |g| For 2 point masses, we know

For real body, must divide into infinitesimal mass elements, dm, find gravity due to each, then find vector sum

It is often convenient to use concept of potential. Potential field is a scalar field from which the vector gravity field can be

found; other examples: elevation-slope, temperature-heat flow

If a (vector) force field is conservative, it may be represented by (the gradient of) a scalar potential.  If a force field has a scalar potential, it is conservative.

Potential field/scalar field example

change in potential between two points is work done to move from point A to point B

since conservative, work done to move from point B to point A is equal and opposite

therefore, work done to go in a closed path is zero; i.e., conservative 

Work done to take a point mass from location r to infinity

gravity is the gradient of potential

the components of gravity, then, are

Escape Velocity

Escape Velocity Escape velocity is the velocity required to launch an object to escape a planet's gravitational force (not just in orbit around the planet).  We have seen that gravitational potential at a point is the work required to move a unit mass from that point to infinity. Potential has units of energy/mass. We found that gravitational potential, U, is given by

where M is the mass of the point mass for which we are finding potential (in this case, mass of the planet, insofar as a planet is spherically symmetric) and R is the distant from the point mass (or from the center of the planet).

To escape a planet's gravitational field, it must have kinetic energy equal to the gravitational potential energy. Kinetic energy of an object, with mass m, is given by

or, per unit mass,

Escape velocity is obtained by setting these equal and solving for V:

Using values for an object starting at the Earth's surface, we get

Finding g from U in spherical

coordinates

Note on signs: defined this way, g will be negative, because it points in the opposite 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 positive number, for convenience.

A note about g, and G!

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

For an arbitrary mass distribution (Cartesian coordinates)

For an arbitrary mass distribution (spherical coordinates)

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

Poisson's and LaPlace's equationsDeriving Poisson's and LaPlace's equationsMass inside the volume.

From Gauss's theorem:

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: the surface presents just as much of its front as its back, so they cancel, or notice that the flux lines which go in one side of the volume bounded by the surface come 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 (where density is zero.)

Applications of Poisson's Equation in Integral Form

From derivation above, we have:

[Consider the equation above "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 imaginary surface ("Gaussian surface") anywhere within the hollow region around the sphere

Since the mass contained within the Gaussian surface is zero,

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

3. Gravity due to an infinite slab of thickness h and density ρ:  Bouguer's Formula

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:

Problems to contemplate:

find gravity {g(r)} inside and outside a homogeneous sphere find gravity {g(r)} outside (r>R) an infinite cylinder with mass per unit

length σ (or, if it makes is easier to visualize, radius R and density ρ)

find gravity {g(r)} inside (r<R) a homogeneous infinite cylinder of radius R with constant density ρ

General Solution to LaPlace's Equation in Spherical Harmonics (Spherical Harmonic Analysis)Elliptical, Parabolic and Hyperbolic PDEs

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

In spherical coordinates, where r is distance from the origin of the coordinate system, is the colatitude, and is azimuth or longitude:

Solutions to LaPlace's equation are called harmonics In spherical coordinates, the solutions would be spherical

harmonics

Example: show that for point mass ( )

Solving LaPlace's Equation

Assume variables are separable: , so

Multiply through by :

Last term on LHS depends only on λ, yet first two do not depend on λ, so last term must be constant (and first two must add up to negative of that constant).

This is of the form (this is probably where the term "harmonics" arises; see below "Simple Harmonic Oscillator problem)

This an ODE, with solution , where m is an integer Going back to the first two terms, we have

Multiply through by :

Again, terms must be independent, so both must be constant:

or which has the solution 

Finally,

This is known as Legendre's Equation, and has solutions of the form

, where are the Associated Legendre Polynomials,

are constants

General comments on differential equations:

All soluble differential equations have a general solution (by definition). The general solution always has undetermined constants (which is why they are general), similar to, if not identical to, constants of

integration To apply the general solution to a particular problem, you have to apply boundary conditions and (if time-dependent) initial

conditions(see Simple Harmonic Oscillator problem)

For the general solution to LaPlace's equation, there are a couple boundary conditions that just make physical sense. The rest are determined by measurements of gravity around the Earth.

The general solution to LaPlace's Equation, then, is:  [here a is mean Earth radius]

 

Like any differential equation, the undetermined coefficients, in this case Cl

m, C'lm, Slm, S'lm (an infinite number of them!), must be determined by

boundary conditions. A few of these are "common sense" boundary conditions; the rest have to be determined by best fit of the various harmonics to the Earth's gravitational field. Since we are continually improving our knowledge of gravity, the values of these constants are being refined.

1. Since a body that is finite in three dimensions (x, y, z) will "look like" a point mass at infinity, the gravity must tend to GM/r2 as r

goes to infinity, so the potential will go to -GM/r. This eliminates the C'lm, S'lm terms, because they depend on r l

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 C00 is

identically equal to GM/r, where G is the Univ..., M is the mass of the 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 and south of the center of mass, and in front and behind the center of mass.  Therefore, the l = 1, m = 0 term must be zero, because it is asymmetrical 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).

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

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

for rotating Earth, might neglect λ dependence, i.e., allow only m = 0 terms:

where are Legendre polynomials

or, for convenience

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

From Lowrie (2.51, p. 63; Lowrie calls the Earth's mass E, instead of M )

Values determined by satellite:

(oblateness) (pear-shapedness)

measurements of Earth's gravity field (primarily by satellite

tracking) show that the biggest effect is due to Earth's rotation and bulge

Finally, we can get g from U by taking the gradient; Just keeping the first 3 terms leads to the International Gravity Formula (IGF; also sometimes called "normal" gravity), which we will present in the next section.

Zonal, Sectoral, Tesseral Components

zonal harmonics: m = 0, no longitudinal variation sectoral harmonics: l = m, no latitudinal variation general case: tesseral components (tessarae: Gr., "tiles")

m  gives the number of nodal planes crossed on front side, E to W

l-m  gives the number of nodal planes crossed from N pole to S pole

Shape of the EarthThis is the approximate shape of a rotating  fluid body, and approximates the shape of the Earth.  It is exactly of the form:

where a is the equatorial radius, and c is the polar radius, and β is latitude. Equatorial radius is 6378.1 km.  Polar radius is 6356.8. [Volumetric mean radius is 6371.0 km.]

International Gravity Formula (IGF)The IGF is a best fit to the Earth's gravity as determined from spherical harmonic analysis of the Earth's potential, using satellites. In these equations, θ is geographic latitude. The IGF (g0) is also commonly referred to as theoretical gravity or normal gravity

First internationally accepted IGF was 1930:

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

The Geodetic Reference System1967 provided the 1967 IGF:

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

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

International Gravity Formula "Calculator"

The GeoidThe Earth's Geoid

Earth's Geoid: The geoid is a representation of the surface of the earth that it would assume if the sea covered the earth, also known as surface of equal gravitational potential, and is essentially mean sea level. Remember, sea level isn't flat! The vertical coordinate, Z (elevation), is referenced to the geoid.

Can be defined as:

the shape a fluid Earth would have if it had exactly the gravity field of the Earth an equipotential surface roughly the sea-level surface - dynamic effects such as waves, and tides, must be

excluded geoid on continents lies below continents - corresponds to level of nearly massless fluid

if narrow channels were cut through continents geoid highs are gravity highs g (vector gravity), or vertical, is perpendicular to the geoid: "What's up?" "Perpendicular

to the geoid." This because: o potential at a point is "work done to take a unit mass from that point to infinity."

Over a density high, this would require starting at a higher point o gravity will be deflected toward a dense body o fluid will be attracted toward a dense body, raising its surface

From scienceworld.wolfram.com:

The shape of an object's gravitational equipotential surface. For the Earth, the reference geoid is

where is the colatitude. 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.

An equipotential map of the Earth is dominated by the variation in gravity (and hence geoid height, or basically the shape of the Earth) caused by the Earth's rotation and subsequent flattening. It would look something like this:

The wiggles you see on the contour lines are actually just gridding/contour artifacts.

Ellipsoid

An ellipsoid is a smooth elliptical model of the earth's surface. X,Y (horizontal coordinates) are referenced to an ellipsoid.

In the past, different regions of the world had adopted "local" versions of the elllipsoid The Clarke 1866 ellipsoid is a predecessor to the GRS80 ellipsoid that was used in

North America and is still the reference geoid on many maps In the the "space age," however, a "universal" system was required GRS80 is currently the most commonly used elliptical model used for the globe,though a new

ellipsoid has recently been developed by the National Geodetic Survey and will likely replace GRS80 for future projects.

Reference Ellipsoids used in Geodesy

Name of ellipsoid semimajor axis a[m]

flattening f = (a-b)/a applied for

Geodetic Reference System 1980 (GRS80)

6 378 137. 1 : 298.25722 World Geodetic System 1984

World Geodetic System 1972 (WGS72) 6 378 135. 1 : 298.26 World Geodetic System 1972

Geodetic Reference System 1967 6 378 160. 1 : 298.25 Australian Datum 1966 South American Datum 1969

Krassovski (1942) 6 378 245. 1 : 298.3 Pulkovo Datum 1942

International (Hayford 1924) 6 378 388. 1 : 297.0 European Datum 1950

Clark (1866) 6 378 206. 1 : 294.98 North American Datum 1927

Bessel (1841) 6 377 397. 1 : 299.15 German DHDN

What does elevation above mean sea level mean?

Sea Level: Frequently Asked Questions and Answers

Why does a GPS measure elevation relative to WGS84, or some other reference ellipsoid, whereas a surveyor's estimate is relative to the geoid?

“My dog isn’t a piglet”. Vladimir Putin’s dog Koni prepares to test Russia’s new GPS system Photo: AP

Geoid Anomaly:  A change in the height of a portion of the geoid compared to its height for a flattened ellipsoid. On Earth, substantial geoid anomalies are found at subduction zones and hotspots. In continental regions, they do not correlate with topography because of isostatic compensation . On both Venus and Mars, however, geoid anomalies are correlated with topography.  © Eric W. Weisstein

Before looking at the geoid, which is dominated by the J2 term, that term is removed, which amounts to removing an ellipsoid of flattening of (as currently determined) 1/298.25:

Below is

Image Name : ww15mgh;      Boundaries : Lat -90N to 90N;     Lon 0E to 360E;Color Scale, Upper (Red) : 85.4 meters and higher;       Color Scale, Lower (Magenta) :-107.0 meters and lower Data Max value : 85.4 meters       Data Min value :-107.0 meters        Illuminated from the : East

This is an image generated from 15'x15' geoid undulations covering the planet Earth. These undulations represent the NIMA/GSFC WGS-84 EGM96 15' Geoid Height File. This file is a global grid of undulations generated from: (a) the EGM96 spherical harmonic coefficients and (b) correction terms that convert pseudo-height anomalies on the ellipsoid to geoid undulations.

This file may be found at: http://164.214.2.59/geospatial/products/GandG/wgs-84/geos.html.  The undulations in this file refer to the WGS-84(G873) reference ellipsoid.  Some interesting features to note about this image are: Even at 15' resolution, some beautiful features of the global geoid are obvious.  The major trench systems have obvious impacts on the geoid, as well as the topography/ ocean boundaries (whose geoid signals closely coincide with the shoreline).

The Hawaiian Island chain may be followed up through its transition into the Emperor Seamounts and toward the western end of the Aleutian Islands.  The structure of seamounts with the Marshall Islands, east of the Mariana Trench, can be seen in the geoid signal in that area.  Finally, the well-known geoid low near the tip of India, and the geoid high over New Guinea stand out, with a great deal of finely detailed structures

mixed in with these broad features.    Map and description from the National Geodetic Survey.

Ocean Geoid

Geoid Over Subduction Zones

The most prominent features on most geoid maps (depending on filtering used) are subduction zones:

Cross-sections across subduction-zone geoid anomalies show an asymmetric anomaly low (trench) and an anomaly high (presence of cold, dense slab in lighter asthenospere):

Fig. 1. (A) Free-air gravity anomaly from satellite altimetry for the Tonga-Kermadec region. (B) Free-air gravity anomaly for 3D dynamic model including a low viscosity region in the wedge. (C) Comparison of topography along east west profiles across the subduction zone at 20, 25 and 30°S (thick/blue) to observed topography (thin/black). Model topography has an arbitrary reference height (here set to zero) therefore, observed topography is adjusted to equal zero at the model boundary. (D) Comparison of model geoid anomalies (thick/blue) with observed along east west profiles. An east west linear ramp is removed from each of the observed and model geoid profiles so that the geoid equals zero at the model boundaries.

Geoid of the United States

GEOID99 is a refined model of the geoid in the United States, which supersedes the previous models GEOID90, GEOID93, and GEOID96. For the conterminous United States (CONUS), GEOID99 heights range from a low of -50.97 meters (magenta) in the Atlantic Ocean to a high of 3.23 meters (red) in the Labrador Strait. However, these geoid heights are only reliable within CONUS due to the limited extents of the data used to compute it. GEOID99 models are also available for Alaska, Hawaii, and Puerto Rico & the U.S. Virgin Islands.

"More than any other data set of the Earth the Geoid shows us the dynamic structure of the Earth's deep interior. The most dramatic feature in the Geoid of North American is the Yellowstone Hot Spot,  believed to be a plume structure rising through the mantle and the main contributor to the Geoid high over Montana. Details of the topographic anomalies of the Western Rockies can be seen superimposed upon this anomaly, although with much less magnitude. The Great San Joaquin Valley of California, formed through the tectonics of the earlier subduction of the Pacific plate by North America is outlined in detail in the Geoid.

Comparison with this feature can be made with those smaller yet similar Geoid lows to the north in Oregon and Washington state. In the midcontinent an ancient rift or suture zone can be seen in sharp outline running from the tip of Lake Superior through Minnesota and continuing to Texas. The Eastern offshore shows some of the oldest portions of the Atlantic Ocean formed some 120 million years ago with its now characteristic Geoid low centered off the Carolinas. Seen also is a deep suture structure running the length of the Hudson River Valley to the opening of the Gulf of Saint Laurence. At the very top of the figure on the right can be seen the outline of the most recently formed feature of Geoid of North America. This is the postglacial Geoid low caused by the depression of the continent under the ice load from the last Ice Age some

20,000 years ago. Because of the viscous nature of the Earth's Mantle this feature will slowly disappear until the end of the next Ice Age when the process will repeat itself again."

By: Allen Joel Anderson Department of Physics University of California

The GRACE Experiment

[From the Jules Verne Voyager:  http://jules.unavco.org/Voyager/Venus?grd=6]

egm96_geoid: The Geoid is that equipotential surface of the Earth gravity field that most closely approximates the mean sea surface. At every point the geoid surface is perpendicular to the local plumb line. It is therefore a natural reference for heights - measured along the plumb line. At the same time, the geoid is the most graphical representation of the Earth gravity field. The geoid surface is described by geoid heights that refer to a suitable Earth reference ellipsoid. Geoid heights are relative small.The minimum of some -106 meter is located at the Indian Ocean. The maximum geoid height is about 85 meter. The figure below shows a global map with geoid heights of the EGM96 gravity field model, computed relative to the GRS80 ellipsoid

From www.gfz-potsdam.de/ news/foto/champ/welcome.html

Die Abweichungen der physikalischen Oberfläche der Erde (Geoid oder 'Normal Null') von einem regelmässigen Ellipsoid, vom Computer mit 15000facher Überhoehung gezeichnet, sind Ausdruck der unregelmässigen Dichte- und Massenverteilung im Erdinnern. Die sich unter dem Einfluss des Erdschwerefeldes ausbildenden Verformungen reichen von -110m im Indischen Ozean bis +90m ueber Südostasien. Die Grossstrukturen dieser Figur der Erde konnten mit dem Mitte 2000 gestarteten deutschen Satelliten CHAMP mit bisher unerreichter Genauigkeit aus Beobachtungen seiner Bahnstörungen ausgemessen werden. Über den Kontinenten ist das Geoid zur besseren Unterscheidung in Graustufen dargestellt.

The GOCE (Gravity field and steady-state Ocean Circulation Explorer) mission will measure high-accuracy gravity gradients and provide a global model of the Earth's gravity field and of the geoid. The geoid (the surface of equal gravitational potential of a hypothetical ocean at rest) serves as the classical reference for all topographical features. The accuracy of its determination is important for surveying and geodesy, and in studies of Earth interior processes, ocean circulation, ice motion and sea-level change.

Credits: ESA

 

WAPGEO_anom_20_270_360.   Free air anomaly map of the Weddel sea.

Geoid of Other Bodies

The geoid, and gravity, can be determined for other planets from satellite data.

Mars

The Burroughs crater on Mars is named in Burroughs' honor.

Venus

Geoid model, derived from Magellan orbit data, spherical harmonic fit (1° resolution).

Moon

Jules Verne Voyager: http://jules.unavco.org

Welcome to the Jules Map Server

To better understand the inter-relationships of geophysical and geological processes, structures, and measurements with high-precision GPS monument data and solutions, the Data Management and Archiving Group has developed an interactive map tool for virtual exploration of Earth and other worlds:

Global gravity maps and the structure of the Earth, 1985, Carl Bowin, The Utility of Regional and Magnetic Anomaly Maps, William J. Hinze, Ed., S.E.G.

Would gravity still be less due to the equatorial bulge (neglecting rotation effect) for a homogeneous planet?

The bulge has 2 effects: it reduces g because one is farther from the center of the planet (free-air effect) but increases g due to the mass of the bulge (Bouguer effect). The "real" Earth has a significant central condensation, i.e., it gets denser toward the center and the bulge alone produces a decrease in gravity approximately equal to the decrease caused by rotation alone.

But what about a homogeneous planet? Assume that, to first order, the planet is spherical. The gravity from the surface outward would be given by

where R is the planet radius and ρ is density. The decrease of g(r) with r (elevation), would be

Evaluated at the planet's surface, r=R, we get

The attraction due to the mass of the bulge can be approximated by computing the gravity of an infinite slab (Bouguer effect) which is given by

Thus the elevation effect is bigger by

 

 

 

 

  

 

Seismology and Earth's InteriorThere are two categories of earthquake waves. Body waves can travel deep into the Earth; Surface waves can only travel very near the surface of the Earth. There are two kinds of body waves, and two kinds of surface waves. As you might imagine, only body waves can give us any information about the deep interior of the Earth.

All earthquakes are relatively shallow, with the deepest at about 700 km depth. An earthquake generates body waves that spread out in all directions, like light from a naked light bulb. Notice in the diagram below that you can think of earthquake waves as moving out like rays (arrows) or as wave fronts (spherical shells). Surface wave rays travel out in all horizontal directions (like the arrows on the top of the block pictured below), like ripples moving out from a pebble dropped into a pond.

All over the surface of the Earth are seismograph stations which can detect all of the waves that arrive at that location. By recognizing what kinds of waves have arrived, exactly when they arrived, and knowing where and when the earthquake occurred (or sometimes the earthquake location and time itself is determined by seismograph stations), we can learn about the deep interior of the Earth. This is because these waves refract (bend) and reflect at boundaries in the Earth.

Strain is dimensionless; Stress has units of force/unit area. 1 Newton/m2 = 1 Pascal = 1 PaBody Waves:

There are two kinds of body waves corresponding to the two fundamental ways you can deform an object: you can squeeze it (or stretch it, which is like "negative squeezing"), or you can shear it. 

P Waves

The diagram on the left above illustrates a P wave. These are also called compressional or longitudinal waves. Material is compressed and stretched in the

horizontal direction, from left to right, and the wave (disturbance) also travels in the horizontal direction. P waves travel faster than any other type of wave. They can travel through fluid or solid materials. Ordinary sound waves in air are P waves.

P comes from primary wave, because they arrive first, but a mnemonic is push-pull wave

P wave velocity depends on a material's "plane wave modulus" and its density:

Where λ is Lamé's constant, μ is shear modulus, K is bulk modulus, and ρ is density. Notice that density is in the denominator, so denser rocks should be slower. However, although the density of rock in the Earth generally increases with depth, the rigidity, as expressed in the various elastic constants, increases even more rapidly with depth. Hence, P wave velocity generally increases with increasing depth.

Since solids, liquids and gasses have a finite bulk modulus, P waves can travel through any of these

S Waves

The diagram on the right above illustrates an S wave. These are also called shear waves. S comes from secondary wave. Material is sheared, so that an imaginary square drawn on the side of the block becomes diamond shaped. The material vibrates up and down (or side to side, in and out of the screen, if the hammer had struck the side of the block instead of the top) but the wave (disturbance) travels in the horizontal direction from left to right. S waves travel more slowly than P waves. They can only travel through solid materials. Plucking a guitar string generates a kind of shear wave; the string vibrates side to side, but the wave travels along the string.

S-wave velocity depends on a material's shear modulus, μ, and density, ρ:

Since fluids (liquids and gasses have zero shear modulus, S waves cannot travel through fluids. However, seismic waves have a period no larger than minutes. Some materials, like the mantle, are solids on that time scale, but not on the time scale of millions of years.

Comparing the velocity expressions, you can see that VP > VS for any material.

For both types of body waves:

P and S waves travel faster in rigid, dense rocks. Rocks generally get more rigid and denser with depth. Generally, though, elastic constants increase more rapidly than density, so the velocity of P and S waves generally increases with depth.

P and S waves are refracted and reflected at boundaries. In the diagram below, the subsurface earthquake location (focus, or

hypocenter ) is shown in yellow. The ray we've shown coming out of the earthquake travels in a straight line in the blue layer. When it reaches the red layer (which might be slower or faster), the ray splits: some of the energy goes into the red layer but is bent (refracted), and some of the energy is reflected back up to the surface. An analogy: When you stand in front of a store window, you can usually see your reflection, proving that some of the light reflects back at you. But people in the store can also see you, so some of the light goes through the glass. [Reflectivity can be calculated from Zoeppritz equations.]

 

Surface Waves waves which travel only along the Earth's surface amplitude decreases exponentially with depth (relative to wavelength) thus, short wavelength have shallow penetration similar to skin effect of electromagnetic waves traveling along

conductive medium (submarine comms)

Historically, recognized as "large" waves, hence designated by "L"

Love Waves

LQ (Quer: German for lateral) horizontally polarized shear waves predicted by A.E.H. Love in 1911 must propagate in a layer, like a layer on a half space

Rayleigh Waves:

LR retrograde elliptical waves predicted in 1885 by Lord Rayleigh require a free surface

VP>VS>VLq>VLr

Reflections, Refractions, Snell's LawMode Conversion

An incident P wave can cause a reflected P and a refracted P, but it can also cause a reflected S and a refracted S; an incident SV can cause a reflected SV and a refracted SV, but also a reflected P and a refracted P. This is known as mode conversion (from P to S, or S to P). See diagrams below under "Snell's Law."

Reflections

Reflections occur when there is an acoustic impedance contrast between two layers:

Sign determines whether polarity reversal occurs:

In upper crust, changes in ρ sometimes small, the reflection coefficient often depends mainly on velocity differences. (Just a rule of thumb.)

Refractions

Refractions occur when velocities differ (if they don't, ray passes through unbent!):

Snell's Law

Snell's law applies to reflections and refractions, even with mode conversion:

In large regions of the Earth, velocity increase gradually with depth, leading to gradual bending of rays; where there are abrupt velocity changes, sharp bending, and reflections, will occur.

These reflected and refracted rays show up as different phases on a seismogram.

Here is a simple one:

Earthquake Seismology and the Interior of the Eartth

The main points about using earthquakes waves to determine the internal structure of the Earth are summarized here, then explained in more detail:

By measuring travel times of earthquake waves to seismograph stations, we can determine velocity structure of Earth

By making graphs of travel time versus distance between earthquakes and seismograph stations, we find

velocity generally increases gradually w/ depth in Earth, due to increasing pressure and rigidity of the rocks

however, there are abrupt velocity changes at certain depths, indicating layering

The 4 major layers in the Earth, from outside in, are the crust, mantle, outer core, and inner core.

The crust is very thin, averaging about 30 km thick in the continents and 5 km thick in the oceans

The mantle is 2900 km thick (almost halfway to the center of the Earth. It is made of dark, dense, ultramafic rock (peridotite).

The outer core is 2300 km thick and is made of a mixture liquid iron (90%) and nickel (10%)

The inner core is at the center of the Earth and has a 1200 km radius; it's made of solid iron (90%) and nickel (10%).

Crust - Mantle Boundary The crust mantle boundary was discovered in 1909 by a seismologist

named Mohorovici (Yugoslav), as a result of his study of an earthquake in Croatia at that time.

He found that, out to about 150 km, the time it took for the earthquake waves to reach each seismograph station was proportional to the distance the station was from the earthquake. He used the familiar time/distance/rate equation (distance = rate*time, or rate = distance/time) to determine that the velocity of the upper crust must be about 6 km/s. In the graph below, this corresponds to the straight line segment on the left, which has a slope of corresponding to 6 km/s.

However, for stations greater than about 150 km from the earthquake, waves did not take as much longer to arrive as if they were traveling at only 6 km/s. In fact, the slope of the second line segment corresponds to a velocity of 8 km/s.

 

Furthermore, Mohorovici figured out that the distance at which the change in slope occurred (about 150 km) can be used to calculate the depth to velocity increase from 6 to 8 km/s. He calculated that the depth to this velocity jump was about 30 km.

We interpret this velocity jump as the crust-mantle boundary, and often refer to it as the Mohorovicic discontinuity, or Moho, for short.

The diagram below shows a cross-section of the crust and mantle, with the earthquake on the left. The triangles on the surface are meant to be seismograph stations at different distances from the earthquake. At short distances, the "direct waves" that travel along the surface will arrive first. However, at greater distances, the waves that travel down to the mantle, and are bent and travel along the top of the mantle at the higher velocity, can arrive before the waves traveling directly along the surface. These refracted waves make up for the extra distance by traveling faster for most of their path. 

 

Seismic refraction experiments like Mohorovici's have been, and still are, being conducted all over the Earth. They indicate that continental

crust is about 35 km thick, but varies greatly from place to place, and oceanic crust is pretty uniformly 5 km thick.

This contour map of the thickness of the Earth's crust was developed from the CRUST 5.1 model. The contour interval is 10 km; we also include the 45 km contour for greater detail on the continents.

Jeffreys-Bullen travel-time curve Another global raypath diagram Seismogram, near Fiji, GOL, 92.2 deg, 6.4 M b

Core - Mantle Boundary

The core-mantle boundary was discovered in 1913 by a seismologist named Gutenberg. Seismologists had noticed that P waves are not recorded at seismograph stations which are from 104o to 140o away from an earthquake (the angle is the angle made by drawing a line from the earthquake to the center of the Earth, and then from there to the seismograph station.

Gutenberg explained this Shadow Zone with a core which slowed and bent P waves

Later, an S wave shadow zone was recognized, meaning no S waves were received at seismographs stations from 104o to 180o from an

earthquake; the S wave shadow zone is caused by the outer core, which is liquid iron/nickel.

Modeling of seismic waves traveling through the Earth allowed seismologists to determine that the core begins at a depth of 2900 km, or in other words, the mantle is 2900 km thick; its composition is probably ultramafic rock (peridotite). This is based on the velocity of the waves, meteorites, mass of the Earth and other lines of evidence.

Inner Core - Outer Core Boundary In 1936, a Swedish seismologist named Inge Lehmann recognized

waves which were reflected from a boundary deep within the Earth. She correctly interpreted this as the outside of the inner core, which is solid iron and nickel.

In the 1960's, nuclear blasts allowed for a more precise determination of the radius of the innner core. U.S.'s nuclear blasts were always at a known spot, and were detonated exactly at a specified time. This eliminated much of the uncertainty seismologists have to deal with with

natural earthquakes, whose precise origin time and location must be worked out by the travel times themselves!

Copyright 2009 J. L. Ahern

Seismic Refraction MethodApplications:

depth of weathering zone (used for statics correction to seismic reflection data), i.e., depth to bedrock

depth of groundwater table depth of basement depth of Moho depth of any faster unit

For our purposes, assume flat (not necessarily horizontal), homogeneous layers. In order to get a head wave, V2>V1!

The critical angle is the incident angle where the head wave begins:

Snell's law for multiple parallel layers

Refracted angle into one layer becomes incident angle into next layer:

Travel-time curve for single horizontal layer on a half-space:

so velocities gotten from reciprocal of the slopes of the direct and refracted segments, and depth gotten from reflected time intercept (or cross-over distance). However, often only first arrivals are recorded:

Single horizontal layer on a half-space, V2>V1:

Alternatively, in terms of Ti2, the intercept time from the second travel-time segment,

Two horizontal layers on a half-space, V3>V2>V1

where the depth to the lower interface is the sum of z1 and z2, where z1 is computed by the single-layer formula above.

Single dipping layer on a half-space, V2>V1:

Example of ambiguity problem: Shooting up-dip gives apparent velocity that is too fast; vice-versa. VA, up-dip velocity, is too fast (shallow slope), VB, down-dip velocity is too slow (steeper dip). Note that, without reversing the profile, could not distinguish from horizontal case. Note also the total travel-time from end to end is same in either direction: reciprocity theorem.

 

                                                                                                                                                                                                                                                                   

Ambiguity Problem! Low Velocity Layers

never get a refracted head-wave from a slow layer underlying a fast layer

eventually get head-wave when faster layer (V>V1) encountered: e.g.,   V2<V1<V3

The travel time curve will look like this (another example of ambiguity):

interpreter assumes a layer-over--half-space model (can't "see" layer 2) t0 (delay in getting down to layer 3 and back) is large because V2 is slow causes layer to be interpreted thicker than it really is

Typical Reversed Seismic Refraction Profile

 

 

Seismic Reflection Method

How Karcher "hand migrated" dipping layer

Top: Monument unveiled in 1971 at Belle Isle (Oklahoma City) on 50th anniversary of first seismic reflection survey by J. C. Karcher. Middle: Two early reflection records from Belle Isle, 1921. Bottom: Karcher's interpretation of same.

uses reflected energy from interfaces between subsurface layers to determine their configuration

reflections recorded as two-way (down and back up) travel times, not depths

fraction of incident energy reflected from interface called reflection coefficient

dependent on acoustic impedance contrast across interface

Polarity of reflected wave depends on sign of reflection coefficient (unchanged polarity means compression remains compression, dilatation remains dilatation)

Hypothetical Rock PropertiesRock VP, km/s ρ, kg/m3 V x ρGranite 5.0 2700 13,500Basalt 5.5 3000 16,500Limestone 6.0 2300 13,800

Sandstone 4.2 2500 10,500

Shale 2.5 2300 5,750

For these hypothetical values, limestone-granite contact will be poor reflector

Simple Zero-offset Reflection Survey

zero offset (distance between source and receiver) single layer on half-space

reflections produce a time-section approximating interface (that even geologists can understand)

travel time:

2 problems: what is V? [We measure t. If we knew V, we could find d -

convert time section to depth section.] energy source is expensive in time and money

solution to both problems: use geophones at different offsets

Seismic Reflection Survey with Offset, Single-fold Coverage

record traces from several geophones spaced away from source (shot) display traces side-by-side, so distance between traces proportional to

geophone spacing display increasing time downward (time approx. proportional depth)

Note that subsurface reflection points have half the spacing of geophones. To get complete "single-fold "coverage of the subsurface, can shoot from either end of geophone spread:

One can also use a "split-spread" arrangement, here with shot at point B, then move half the geophones forward and shoot at C:

The next two figures show recording-truck signal check for 36-channel split-spread layout:

 

The travel time for the primary reflection (first layer) where geophone offset = x, thickness d, velocity V

this is a hyperbola:

for multiple geophones, seismic traces look like:

direct ray is straight line, per rate equation reflections from 1st and 2nd reflectors are not flat; reflections are

hyperbolas because of normal moveout (NMO): reflection time increases with x,

nonlinearly normal move-out (NMO): the difference in reflection travel-times from a

horizontal reflecting surface due to variations in the source-geophone distance

can correct for this using NMO correction, so reflections are flat because deeper reflectors produce flatter parabolas, NMO is less:

given that :

the NMO is just t - t0, where t0 is simply 2d/V (zero-offset 2-way travel time)

but we don't know d or V

however,

  so, plot t2 vs. x2:

measure slope to get V intercept gives d

Multiple Layers

foregoing assumed a straight-line path from source to reflector to receiver

with multiple layers with different velocities, this clearly does not hold (actual path compared with straight-line assumption):

however, if depths are large compared to total geophone spread, error can be small

Green Method

assuming straight-line paths, one can still just use a x2-t2 plot to estimate velocities and depths

however, there is a more accurate method:

The Dix Equation

uses special velocity called VRMS still assumes nearly vertical incidence/straight line raypaths given n horizontal beds, and δ is the one-way vertical travel-time

through bed i, Dix equation states:

use x2-t2 plot to determine RMS velocities to each layer then can get interval layer velocities and thicknesses replace velocity terms by VRMS

it can be shown that Dix's equation can be solved for the individual interval velocities

in fact, the following is sometimes referred to as Dix's equation:

thicknesses can then be easily determined:

Velocity Scans - another way to get stacking velocity:

Signal Summing; Stacking

As seismic energy moves away from the source, there is a decrease in signal strength as the energy spreads out.

This causes energy to decrease by E = E0/(2πr2). Since energy is proportional to the square of the amplitude, the signal

amplitude drops off like 1/r. In addition, since rocks are not truly elastic (anelastic), some energy is

lost to heat with every cycle, leading to an exponential loss of energy. Higher frequencies go though more cycles to a given depth (shorter

wavelength), so high frequency energy is lost with depth Taking both of these effects into consideration, we have:

reflections from significant depth have amplitudes that may be well below the noise level

summing and stacking adds (coherent) signals and (random) noise improves signal-to-noise ratio (S/N, or S/(S+N))

summing n times increases signal by n summing n times increases noise by square root of n

Example: reshoot a line 36 times signal increases by 36 noise increases by 6 (square root) S/N improves by 36/6 = 6

different means of improving S/N:

geophone groups

"Geophones are rarely used singly. Normally several (as many as 20 or more) are electrically connected to each other in a group in such a way that the outputs of the individual phones are effectively summed. The information from each group must be transmitted via cables to the recording truck. In modern land recording with 48, 96, or more group recordings, the cables are long and heavy and often add noise to the recording, especially in the presence of powerlines or water." - Dobrin and Savit, Introduction to Geophysical Prospecting, 4th ed.

"geophone" is actually a group of geophones "tied together" in a geophone group

signals in parallel, fed into one channel of system signal usually has small incident angle, reaches all geophones together

(coherent) surface noise sweeps across geophones, tends to cancel

multiple shots

dynamite in hole vibroseis trucks: multiple trucks, all in sync; shake several times multiple hammer blows, shotgun blasts, etc.

multi-fold coverage

Example: 4 geophones (channels); move shot and geophones one geophone spacing and reshoot:

note that subsurface reflection points twice as closely spaced as geophones

had we moved shot (array) 2 geophone spacings, only get single-fold coverage

try this with different numbers of geophones and shot spacings to find a simple formula to calculate fold-coverage (coverage = f(# geophones, # geophone spacings moved between shots)

Rock Velocities

P, S velocities of various rocks (See Sydney Clark's GSA Memoir for much more)

velocity vs. density velocity vs. density, part 2 velocity vs. rock age for sandstones and shales

Data Collection

Source

Geophones

Geophones (~ $100 each) have a typical natural frequency of 10 Hz Response is relative good over a range from about 2 Hz to 100 Hz

Recording Digitally

Analog signal from phones (continuous voltage vs. time)

sampled (typically every 2 msec) by an A-to-D (A/D) converter originally integer recording

example: 16-bit recording; 216 = 65,536, or -32767 to +32768 dynamic range of about 4.5 orders of magnitude poor resolution at low amplitudes

floating pointing recording single precision, 4 bytes, 6 digits of resolution, dynamic range

10+/-32

Processing Steps

At one time, greatest computing power was owned by government (mostly DOD)

Petroleum companies ranked second most seismic reflection processing is computer intensive, but requires

intelligent "operator" input at many steps in the process

AGC: automatic gain control

early-arriving reflections may be orders of magnitude larger in amplitude than later ones

AGC looks at average amplitude in a sliding time window and boosts (or attenuates) amplitude to a constant value over that window

AGC causes loss of true amplitude information modern floating-point recording allows full amplitude information to be

retained retaining "relative, true amplitude" done with linear or quadratic

increase of gain with time:

Filtering

remove or attenuate certain frequencies to reduce noise and improve S/N

notch filter common at 60 Hz filtering often done with FFT filters must be "ramped" to avoid ringing (Gibbs phemonemon)

Statics Removal

refraction statics - requires reversed profile up-hole shooting

vertical velocity distribution near the surface determined by shooting up the hole" (Geophysical Services, Inc.):

automatic statics

Migration

Synthetic Seismograms

Directions Seismic Reflection is Heading

Percentage of seismic activity involving various techniques. (Data from SEG annual Geophysical Activity Reports, pre-1981 data are for U.S. activity, post-1980 for worldwide activity, 3-D data from Dutt, 1992, adjusted according to judgment expressed in Goodfellow, 1991.)

Seismic Attributes

reflections are not only information available in seismic data already seen value of preserving "relative, true" amplitude preservation and display of velocity data can reveal info otherwise

missed:

Conventional B&W section on which carbonate bank would be missed:

Color display in which colors are keyed to interval velocity estimates (1000 ft/s increments):

Close-up of carbonate bank sequence seen above:

3-D Seismic Reflection

3-D representation ("data cube") migration out of plane of section (side-swipe) geology is, after, 3-D much more expensive! (~n2)

3-D sesimic time slices at time ranging from 1060 ms to 1260 ms:

3D visualization (caves, virtual reality, etc.) wavelet processing

Summary of Processing Steps