Velocity Structure of the Earth

Question:what are the sources of heat energy within the earth?

(J. Louie) Above is a simplified view of the composition and state of regions within the Earth. Note how the regions are arranged in radial shells. Compositions and properties vary vetween shells a great deal, while relatively little within any one shell. While planetary physics, the properties of the geomagnetic field, Earth's gravity field, tides, geochemistry, and geology all contribute to knowledge of the Earth's interior, defining and characterizing the layers of the Earth has basically been the business of earthquake seismologists.

(from *Kearey & Vine*, copyright Blackwell Sci. Publ.)
Timing of the arrivals of seismic waves is, of course, most sensitive to
the velocity property of the materials encountered.
While other properties, such as incompressibility, rigidity, and density
may be *inferred* from velocities, only velocities can be measured
accurately and directly.

Note how the transitions between layers show both *gradients* and
*discontinuities*:

- Discontinuities arise at changes in
*composition*. - Steep gradients arise at changes to denser, tougher
*phases*(having the same chemical composition), or perhaps at changes in*temperature*(convection boundary layers) or composition. - Shallow gradients arise where the same phases and assemblages of minerals undergo velocity increase with pressure concurrently with velocity decrease with temperature, as depth increases. A slow rise in velocity results.

Depth, km Name

5144Lehman- Fe solid against FeO, FeS fluid 2885Gutenberg- fluid FeO, FeS against (Mg, Fe) silicates, velocity decrease, density increase 2870D''- thin, mixing of mantle and core material? 670"670 km"- worldwide, no earthquakes deeper, debates over whether a composition, phase, or viscosity change 400"400 km"- worldwide, structure more variable above, phase change to spinels 50-200LVZ- really a couple of gradients, regionally variable 4-55Moho(mo-ho-RHO-vi-chich) - sharp compositional change to crust, tectonically active? 5-30Conrad- mafic to felsic crust, often absent

(original
image from the Exploratorium;
used by permission)
The above discontinuities were discovered by Lehman, Gutenberg, Mohorovicic,
and Conrad by virtue of their ability to *refract* or bend seismic
waves, just as a prism bends light waves.

Question:state Snell's Law of refraction in terms of refractive index, and in terms of velocity.

(J. Louie)
Beno Guterberg located the core-mantle boundary from the P-wave shadow
zone, where P-waves are bent away from the boundary, and from the larger
S-wave shadow zone, showing that the core is fluid.

(J. Louie) Just by knowing the delta degree angle of the onset of the S-wave shadow zone, and the radius of the Earth, you can make a simple estimate of the radius of the core.

Question:knowing the true core radius, how much do velocity changes in the mantle bend S waves up?

In a constant velocity medium, there are no refractions, and the t-x plot
shows a straight line through the origin, with a slope of the inverse of
the velocity.

A discontinuity gives you the original direct phase as well as a refraction
(an line offset from the origin) and a reflection (a hyperbola).
There is also a wave transmitted through the discontinuity that appears
elsewhere ...

(from *Kearey & Vine*, copyright Blackwell Sci. Publ.)
such as where a deeper discontinuity interacts with it further.

Question:How many reflections are not shown above?

A simple velocity gradient produces a refraction that takes a circular path, having a hyperbolic sine (or Error Function) shape in the t-x plot.

A gradient discontinuity produces hybrid reflection/refractions, forming
a *triplication* pattern in t-x, since some observation distances
will note 3 arrivals from the discontinuity: a direct wave, a refraction,
and a reflection.

The cut off, of the triplication's back branches at B and C, are useful in estimating the steepness of the gradient, which together with experimental data can determine whether the discontinuity is a compositional or a phase change.

In her thesis Maryanne Walck used these data and synthetics to control the
steepness of the gradients at the 400 and 670 km discontinuities below
the Gulf of California. The t-x diagrams have been skewed by a reduced
time t - t * 10 (km/s), for presentation, so velocities less than 10 km/s
tilt left, and more than 10 km/s tilt right. Walck estimated the 400 km
discontinuity to be a shallow-gradient phase change, while the 670 km
discontinuity had the steep-gradient character of a compositional change.

Question:Identify a 400 km back branch and a 670 km back branch in the data above.

The above methods assume a purely radial or layered symmetry, with no lateral discontinuities. Repeated, localized experiments can show general changes, as in the thickness of the crust, as long as the experiments did not intersect any lateral transition.

(copyrighted figure)
This map of velocity just below the Moho was contoured from many
localized experiments around the 48 states.
The higher velocities below the Sierras are an artifact of the experiments'
spanning a lateral discontinuity.

Question:what is the significance of the low Pn velocities below the Great Basin?

Doctors in the analog age used to x-ray a patient from several directions, arrange the films around a disk of ground glass, and shine lights through them from the outside to back-project the shadows into the glass.Tomographyis the reconstruction of an image from its projections, or shadows.

A tomographic study starts by dividing the region (here a map) into blocks. Number each block with a block numberNote:you won't need to know any of the fancy matrix manipulations below to pass this class. However, you will be tested later on vector algebra, after we go through it.

With many sources and many receivers, the number of rays will be
(#sources)(#receivers) = nr . Each ray, numbered **r**, has a
travel time **t**:
.
**l**rb is the length of ray **r** in block **b**.

Of course the length of any one ray in most blocks is zero. For a couple
of rays, the time summation looks like:

We can write the summation above in matrix form:

or .
The ray length matrix **L** has dimensions **nr** by **nb**,
and is very sparse.

The matrix equation
is a statement
of the *forward* problem,
or how to model synthetic travel times when given a velocity model input.
(Arbitrarily numbering our time data and slowness model values and
packing them into a vector takes some getting used to, but makes the
math a lot simpler.)
In geophysics we are given data input as a set of travel times, and we wish
to solve for an estimated velocity model.

Suppose we could somehow invert **L**:

We can likely at least estimate the ray paths, so we have no problem finding
the ray length matrix **L**.
Then we could determine the slowness model from the time data.

Since our number of observations never matches the number of blocks, andQuestion:what's in the Identity matrixI?

Then the problem reduces to finding the inverse of the square matrix to get the slowness solution

However, a typical tomography problem will be characterizing a region 100 by 100 blocks in size, or having 10,000 total blocks. This means that will be a 10,000 by 10,000 matrix, with 100 million elements. Unless you have a very large supercomputer at your disposal, it is extremely difficult to invert such a large matrix with gaussian elimination or singular-value decomposition.Question:why are these called the normal equations?

The secret of tomography is to make the initial assumption that only theQuestion:even if your computer has a large amount of virtual memory, it won't help you here. Why not?

Inverting

SubstitutingQuestion:what in general would be a way to improve upon a first-order solution? Think, for example about the steps in a long division.

The solution shows that we are just back-projecting each ray onto the model, one at a time, and summing the slownesses into each block. With a total ray length L, we are giving each block hit the average slowness of the ray

We can solve large tomography problems on any computer, since we only need two bins of storage for each block (for the 100x100 block model, maybe just 80 kbytes) and we process the rays one at a time. We never have to keep a huge matrix in memory.

Tomographic solutions do have some serious problems. Most important is a tendency for anomalies to be drawn out in the direction of most of the rays passing through a region. In geophysics it is very difficult to achieve an even and isotropic (omni-directional) ray coverage.

Above is a tomographic inversion result for a section of the Earth's
mantle, at the equator
(Original 0.3 Mbyte
PostScript file from Harvard, used
by permission).
Cool colors represent represent positive
deviations of velocity from the radially-symmetric average at that
depth, and warm colors represent negative velocity deviations.
Since the mantle has a nearly constant composition, velocity deviations
are thought to be due to differences in temperature, with cool colors
for fast cold mantle and warm colors for slow warm mantle.
The dashed circle is the 670 km discontinuity; plate boundaries are
yellow on the index map at center.

Warm, viscous parts of the mantle are less dense than their surroundings and
will rise buoyantly over geologic time (as fast as your fingernails grow).
Relatively cold mantle will sink.
Driven by heat escaping the core, and cooled in the oceans, the mantle will
circulate or *convect*, just like a boiling pot.
A complete turnover takes hundreds of millions of years.
The interiors of all planet-sized bodies must be actively convecting,
to release their heat of formation.
Any planet with a radius over ~1500 km cannot conduct its internal heat
away within the age of the universe, so it must convect viscously to
release its heat, or it would melt and then convect as a fluid.

(3.3 Mbyte printable PostScript file
from Harvard, used
by permission)
A 3-d view of a mercator projection of the mantle, with orange surfaces
surrounding warm blobs of mantle, which should be rising plumes.

Question:what areas are above rising mantle plumes?

(3.3 Mbyte printable PostScript file from Harvard, used by permission) A 3-d view of a mercator projection of the mantle, with blue surfaces surrounding cold blobs of mantle, which should be sinking slabs.

Question:what areas are producing sinking mantle slabs?

Question:do you see any evidence in the above results for or against two layers of convection in the mantle?