GEO 5210 โ Seismology I L14-1

Lecture 14 โ Seismic Inversion Part I

1. A brief introduction to Inverse Theory

Inverse theory in seismology has many applications. Some of the most important are:

1) Earthquake location

2) Determining 1D velocity and density structure of the Earth

3) Seismic tomography โ i.e., 3D velocity variations.

Many problems in the physical sciences can be written in the form of a linear equation:

๐๐๐๐=๐๐

Where we define:

A = model/data kernel. This provides the relationship between model parameters and data.

m = model vector. Contains the unknown model parameters.

d = data vector. Contains our observations.

Example: Straight line fitting

A simple example would be temperature measurements (Ti) made as a function of depth (zi) in the Earth.

Looking at these data we see what looks to be a linear increase of temperature with depth. So, letโs try and describe our ith temperature measurement as follows:

๐๐๐๐=๐๐+๐๐๐ง๐ง๐๐

That is, we want to find the equation of a straight line that best fits our observations. We can write this in matrix form:

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โฃโขโขโขโก๐๐1๐๐2๐๐3๐๐4๐๐5โฆโฅโฅโฅโค=โฃโขโขโขโก11111๐ง๐ง1๐ง๐ง2๐ง๐ง3๐ง๐ง4๐ง๐ง5โฆโฅโฅโฅโค๔๐๐๐๐๔

Where our data vector (d) contains our temperature measurements (T1, T2, โฆ, T5), our model vector (m) contains the unknowns (our slope โ b and intercept โ a), and the model kernel (A) shows the relationship we are assuming between our data and unknowns.

Just multiplying out the first term of our matrix above we can see that this represents the relationship we are looking for:

๐๐1=๐๐+๐๐๐ง๐ง1

If the system above was evenly determined (i.e., we had the same number of equations and unknowns) then we could easily find the solution of ๐๐๐๐=๐๐ by just solving for m. That is:

๐ฆ =๐๐โ1๐๐

We can see why this is called inverse theory, because to find our solutions (our model parameters) we have to find the inverse of the model kernel.

But, our example problem had 2 unknowns and 5 observations. This is often the case that we encounter, i.e., we usually have more observations than unknowns โ this is referred to as an overdetermined system.

Overdetermined Systems

Recall that a matrix A is invertible if there exists a matrix B such that: AB = BA = I. Non-square matrices are not invertible. Hence, in our example above we have a 5 ร 2 matrix, which is not square and not directly invertible.

Traditionally, what is done is to look for the solution that provides the best fit in the least squares sense. Consider the next image:

GEO 5210 โ Seismology I L14-3

Here we show our observed temperature readings (red circles) and a possible line fit to these observations. The black cross shows the predicted (model) value for this line at each depth. What we calculate is the residual. Where we define the residual = observed value โ model value.

The least-squares solution provides the best-fit where we define the best-fit as the line for which the sum of the squared residuals has its least value. That is, we seek to minimize:

๐๐=ฮฃ๐๐๐๐2๐๐๐๐=1

Where ri is the ith residual.

We set up our equations as:

๐๐๐๐=๐๐

The solution by least squares can be determined from multiplying each side of the above equation by the transpose of A: ๐๐๐ก๐ก (see your linear algebra text for proof).

โ๐๐๐ก๐ก๐๐๐๐=๐๐๐ก๐ก๐๐

Now we solve for m:

โ๐ฆ =(๐๐๐ก๐ก๐๐)โ๐๐๐๐๐ก๐ก๐๐

Note that in our example A is a 5 ร 2 matrix. So, At is a 2 ร 5 matrix. So, multiplying ๐๐๐ก๐ก๐๐ we get a 2 ร 2 matrix, which is square, and possibly invertible.

Underdetermined Systems

Underdetermined systems are also possible. That is, we have more unknowns than equations. If we try the least squares solution we find that ๐๐๐ก๐ก๐๐ is singular (i.e., it has no inverse). There are an infinite number of solutions. That is, if we try to fit a straight line to a single data point, any line will fit through that data point:

GEO 5210 โ Seismology I L14-4

This is unfortunate because sometimes parts of problems we are trying to solve may be underdetermined.

The traditional approach to handling this is to look for the simplest solution. That is, we search for the model values that have the shortest length in the least squares sense. Specifically we seek to minimize:

๐๐=ฮฃ๐๐๐๐2๐๐

In this case, the solution is:

๐ฆ =๐๐๐ก๐ก(๐๐๐๐๐ก๐ก)โ1๐๐

To make this clear, letโs take a look at an example. Suppose we have just one measurement:

T1 = 5ยฐ at z1 = 10 m.

So, we set up our matrix equation as:

๐๐๐๐=๐๐

โ[1๐ง๐ง1]๔๐๐๐๐๔=[๐๐1]

โ[110]๔๐๐๐๐๔=[5]

Solving ๐ฆ =๐๐๐ก๐ก(๐๐๐๐๐ก๐ก)โ1๐๐ we get:

โ๔๐๐๐๐๔=๔110๔๔[110]๔110๔๔โ1[5]

โ๔๐๐๐๐๔=๔110๔(101)โ1[5]

โ๔๐๐๐๐๔=

๔110๔๔1101๔[

5]

โ๔๐๐๐๐๔=๔510150101๔

GEO 5210 โ Seismology I L14-5

โ๐๐=5101=0.0495 (the intercept)

โ๐๐=50101=0.495 (the slope)

And our solution length is:

๐๐=ฮฃ๐๐๐๐2๐๐=๐๐2+๐๐2=.247

So, we get a straight line passing through our point that looks like:

But is this solution really of any interest to us?

The problem in geophysics is that we often encounter cases where parts of our problem are over-determined and other parts are under-determined. So, typically we try to seek a compromise between minimizing the prediction error (the residuals) and the solution length.

Damped Least Squares

The damped least squares solution is:

๐ฆ =(๐๐๐ก๐ก๐๐+ฮป2๐๐)โ1๐๐๐ก๐ก๐๐

Where ฮป is a damping factor.

We try to choose a value of ฮป that minimizes both the prediction error and solution length.

โข If ฮป2 is large ๏ the emphasis is on minimizing the solution length.

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The effect of a large damping factor is to produce a smooth model (i.e., we have a small model length) but not necessarily a great fit to the data.

โข If ฮป2 is small ๏ the emphasis is on minimizing the residuals (prediction error).

The effect of a small damping factor is have very small residuals but a very rough model (and a very large solution length).

We typically try to find a balance between these two end-member cases. For example, we can try a range of values for ฮป and produce a trade-off curve as shown below. Here we try to find the optimal solution as a balance between model roughness and model misfit. Calculating model roughness can be done in a variety of ways, for example by looking at the 2nd derivative of the model values.

GEO 5210 โ Seismology I L14-7

But, one must always be cautious of these solutions. The optimal model doesnโt necessarily imply it is really the Earth we are looking at.

The figure below is an example of the effect of damping in a seismic tomography model. This is from the S-wave tomography model S20RTS by Jeroen Ritsema. It shows a cross-section through the Earth (cutting through the central Pacific LLSVP). Here we can see the effect that choosing a different damping parameter can have on the resulting model. On the left we have a heavily damped model which gives us a very smooth model. Whereas on the right we have less damping that results in a much rougher model and stronger perturbations in velocity.

2. Tau-P Inversion

Previously we noted that travel-time vs. distance or ray parameter vs. distance plots do not produce singularly valued functions. To get around this limitation we generate a new function called a Tau-p or ฯ(p) curve.

GEO 5210 โ Seismology I L14-8

Setting up the inverse problem:

Recall from a previous lecture that we defined Tau(p) as follows:

๐๐(๐๐)=2๔ถฑ(๐ข๐ข2โ๐๐2)12 ๐๐๐ ๐ง๐ง๐๐0

We can write this instead for a stack of discrete layers:

๐๐๔ตซ๐๐๐๐๔ตฏ=2๔ท๔ตซ๐ข๐ข๐๐2โ๐๐๐๐2๔ตฏ12๐๐๐๐=1โ๐๐; ๐ข๐ข๐๐>๐๐๐๐

That is, we have measured ฯ at a number of discrete ray parameters (p1, p2, p3, โฆ, pm). And we decide to parameterize our Earth model as a series of n-layers, each with thickness h:

Letโs think about this problem in the sense of straight-line fitting segments of our travel-time vs. distance curve:

GEO 5210 โ Seismology I L14-9

That is, what if we approximate the real travel-time curve (blue), by a series of straight line segments (red segments). Each one of these segments has a ray parameter p, and letโs assume that the slowness is u. This is strictly only true if we are looking at refracted arrivals (headwaves โ see the aside at the end of this lecture), but is a good first approximation in any case, and is true in the limit that our straight line segment lengths go to zero.

Letโs consider a few values of this equation.

๐๐๔ตซ๐๐๐๐๔ตฏ=2๔ท๔ตซ๐ข๐ข๐๐2โ๐๐๐๐2๔ตฏ12๐๐๐๐=1โ๐๐; ๐ข๐ข๐๐>๐๐๐๐

j = 1

๐๐(๐๐1)=2๔ท๔ตซ๐ข๐ข๐๐2โ๐๐12๔ตฏ12๐๐๐๐=1โ๐๐

โ๐๐(๐๐1)=2โ1(๐ข๐ข12โ๐๐12)12+2

โ

2(๐ข๐ข22โ๐๐12)12+โฏ

Recall that in our setup we assume that u1 = p1, u2 = p2, etc. Hence our first term (๐ข๐ข12โ๐๐12)=0. Since we have increasing velocity with depth, that means that u2 < u1. And u1 = p1, so that u2 < p1. Thus, our second term (๐ข๐ข22โ๐๐12)12 is undefined (hence, the condition that ๐ข๐ข๐๐>๐๐๐๐).

So, ๐๐(๐๐1)=0. This makes sense since our first straight-line segment passes through the origin.

j = 2

๐๐(๐๐2)=2โ1(๐ข๐ข12โ๐๐22)12+2

โ2(๐ข๐ข22โ๐๐22)12

The second term is zero here, so:

โ๐๐(๐๐2)=2โ1(๐ข๐ข12โ๐๐22)12

j = 3

Just considering the non-zero terms:

๐๐(๐๐3)=2โ1(๐ข๐ข12โ๐๐32)12+2

โ

2(๐ข๐ข22โ๐๐32)12

All terms

So, it should be obvious by now that we can write this in the form of a matrix:

GEO 5210 โ Seismology I L14-10

๔ตฆ๐๐(๐๐2)๐๐(๐๐3)โฎ๐๐(๐๐๐๐)๔ตช=โฃโขโขโขโขโก2(๐ข๐ข12โ๐๐22)1200โฆ2(๐ข๐ข12โ๐๐32)122(๐ข๐ข22โ๐๐32)120โฆโฎ2(๐ข๐ข12โ๐๐๐๐2)12โฎ2(๐ข๐ข22โ๐๐๐๐2)12โฑโฎโฆ2(๐ข๐ข๐๐2โ๐๐๐๐2)12โฆโฅโฅโฅโฅโค๔ตฆโ1โ2โฎโ๐๐๔ตช

Which is a matrix equation of the form:

๐๐=๐๐๐๐

Which can be inverted for the layer thicknesses (h).

3. 1D Seismic Velocity Models

The first 1-D models of Earthโs seismic velocity structure were set up by Jeffreys and Bullen. Their 1940 JB model consisted of 7 layers labeled A through G as follows:

Layer

Deepest Depth (km)

Region

A

33

Crust

B

413

Upper Mantle

C

984

Mantle Transition

D

2898

Lower Mantle

E

4982

Outer Core

F

5121

Core Transition

G

6371

Inner Core

The layer names A through G are no longer used. But, it is interesting to note that early on the lowermost part of the D layer was observed to be different, hence the D layer was split into the D’ and Dสบ layers. Dสบ is still used for the lowermost part of the mantle. Another interesting feature was the F-layer, or the Core transition zone. This was dropped in future models, but now researchers are putting forth further evidence that there is a transition region from the outer- to inner- core, and hence the F-layer is making a comeback.

More recent developments are:

1981 โ PREM (Preliminary Reference Earth Model) โ by Dziewonski and Anderson.

Note that these models typically invert for seismic velocities as polynomial functions in different bands of the Earth as opposed to a stack of layers as we discussed above.

GEO 5210 โ Seismology I L14-11

PREM model.

1991 โ IASP91 (International Association of Seismology and the Physics of the Earthโs Interior) โ by Kennett and Engdahl.

This model primarily improves the seismic velocities at the mantle transition zone.

1995 โ AK135 โ by Kennett, Engdahl, and Buland.

The main difference in model AK135 from IASP91 is in the velocities near the boundary of the inner core.

The next figure shows the difference between the JB40 model and the IASP91 model. As can be seen the variations arenโt that dramatic.

GEO 5210 โ Seismology I L14-12

From Stein & Wysession.

The Moon

The only other planetary body for which we have 1D seismic velocity profiles is the moon. Recall that we installed seismometers on the moon during the Apollo missions:

Apollo 12 Alsep (Apollo Lunar Surface Experiments Package)

GEO 5210 โ Seismology I L14-13

Several competing models currently exist. But, one of the best models is by Nakamura 1983. Note that these models do not resolve structure down to the lunar core.

Model Nakamura 1983. Plot by Yao Yao (green โ S-wave velocity; red โ P-wave velocity)

4. Density Structure

Models like PREM also include density as a function of depth. How is this determined? The answer can be seen by looking at the Adams-Williamson equation.

Assume that the Earth is made of infinitesimally thin spherical shells with uniform properties.

We know at depth that pressure (P) is given by:

๐๐=๐๐๐๐โ

Where,

g = gravitational acceleration.

ฯ = density

h = depth

If we consider a change in pressure ฮP with increasing depth ฮh:

ฮ๐๐=๐๐๐๐ฮโ

โฮ๐๐ฮโ=๐๐๐๐

GEO 5210 โ Seismology I L14-14

And, in the limit ฮโโถ0

โd๐๐dโ=๐๐๐๐

That is the change in pressure with increasing depth is +ฯg. Going the other way, with increasing radius:

โ๐๐๐ ๐๐๐ =โ๐ (๐๐)๐๐(๐๐)

That is, the pressure decreases with increasing radius.

We also know that the acceleration of gravity at radius r is (just equate Newtonโs second law with Newtonโs law of universal gravitation):

๐ (๐๐)=๐บ๐บ๐๐๐๐๐๐2

Where,

Mr = Mass within radius r.

Thus,

โ๐๐๐ ๐๐๐ =โ๐บ๐บ๐๐๐๐๐๐2๐๐(๐๐)

From the chain rule:

๐๐๐ ๐๐๐ =๐๐๐ ๐๐๐ ๐๐๐ ๐๐๐

โ๐๐๐ ๐๐๐ =โ๐บ๐บ๐๐๐๐๐๐(๐๐)๐๐2๐๐๐ ๐๐๐ (Eqn 1)

Recall that bulk modulus (k) (or Incompressibility) is a materials ability to resist being compressed under a pressure equal in all directions.

Bulk modulus = k VVPโฮฮ=,

Where P is the Pressure (Force/Area) applied to a volume of material given by V. Graphically, if we apply an equal Pressure (P) on all sidesโฆ

GEO 5210 โ Seismology I L14-15

Then the box deforms as followsโฆ

Where the dashed lines show the volume of the original, uncompressed, box.

For small volumes:

๐๐=๐๐๐ ๐๐๐๐โ๐ ๔ต; (๐ธ๐ธ๐ธ๐ธ๐ธ๐ธ 2)

Noting that we have a minus sign so that with increasing pressure we get a smaller volume.

Since we know that ๐๐=๐๐/๐ we can write:

๐๐๐ ๐๐๐๐=๐๐๐๐๐๐๔๐๐๐ ๔

Carrying out the derivative:

โ๐๐๐ ๐๐๐๐=โ๐๐๐ 2

And substituting back in ๐๐=๐๐/๐ :

โ๐๐๐ ๐๐๐๐=โ๐๐๐

GEO 5210 โ Seismology I L14-16

โ๐๐๐๐๐ =โ๐๐๐ ๐๐

Now, if we substitute this into Eqn 2:

โ๐๐=๐๐๐ ๐๐๐๐โ๐ ๔ต=๐๐๐ ๐๐๐ ๐๐=๐๐๐๐๐ ๐๐๐

โ๐๐๐ ๐๐๐ =๐๐๐๐

And substituting this back into Eqn 1:

โ๐๐๐ ๐๐๐ =โ๐บ๐บ๐๐๐๐๐๐(๐๐)๐๐2๐๐๐ ๐๐๐ =โ๐บ๐บ๐๐๐๐๐๐(๐๐)๐๐2 ๐๐(๐๐)๐๐

Now, if we recall our seismic wave speeds:

P-wave velocity = ๐ผ๐ผ=๔ถจ๐๐+43๐๐๐๐

S-wave velocity = ๐ฝ๐ฝ=๔ถง๐๐๐๐

โ๐๐=๐ฝ๐ฝ2๐๐

โ๐ผ๐ผ=๔ถจ๐๐+43๐๐๐๐=๔ถจ๐๐+43๐ฝ๐ฝ2๐๐๐๐

โ๐ผ๐ผ2=๐๐๐๐+4๐ฝ๐ฝ23

Rearranging:

โ๐๐๐๐=๐ผ๐ผ2โ4๐ฝ๐ฝ23

โ๐๐๐๐=1๐ผ๐ผ2โ43๐ฝ๐ฝ2

And substituting back into our previous equation for the rate of change of density with respect to radius:

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โ๐๐๐ ๐๐๐ = โ๐บ๐บ๐๐๐๐๐๐(๐๐)๐๐2 ๐๐(๐๐)๐๐=โ๐บ๐บ๐๐๐๐๐๐(๐๐)๐๐21๐ผ๐ผ2โ43๐ฝ๐ฝ2

If we let ๐๐=๐ผ๐ผ2โ43๐ฝ๐ฝ2, then we arrive at the Adams-Williamson equation:

๐๐๐ ๐๐๐ = โ๐บ๐บ๐๐๐๐๐๐(๐๐)๐๐2๐๐

This equation gives us the density gradient with respect to radius. Unfortunately, density is also on the RHS of this equation, which is what we are interested in solving for. We already know the seismic velocities however in great detail (see previous lecture!). So, how do we solve this problem?

What we do is an iterative approach. That is, we start at the Earthโs surface at a known density:

๐๐(6371 ๐๐๐ )=3300๐๐๐ ๐๐3โ

We know the Mass of the Earth (Mr) and the seismic wave speeds, so we can now calculate the gradient ๐๐๐ ๐๐๐ . Hence, with this gradient we can linearly extrapolate the density to a new smaller radius:

Now, with this new density, we can calculate a new Mr at the new radius, and repeat. Hence, it is an iterative process. Remember, we also have important constraints that must be satisfied:

โข Total Mass of the Earth: ME = 5.974ร1024 kg.

โข Mass at each successive level: ๐๐๐๐=๐๐๐ธ๐ธโ4๐๐โซ๐๐(๐๐)๐๐2๐๐๐ ๐
๐
๐ธ๐ธ๐๐

โข Moment of inertia constant: ๐ผ๐ผ๐๐๐๐2=0.3307

There are a couple problems with direct implementation of this approach that also need to be accounted for.

GEO 5210 โ Seismology I L14-18

โข This assumes adiabatic compression. But, convection is happening in the mantle so an extra term accounting for convection must be included.

โข This approach also assumes that the composition is constant throughout the Earth. This is readily accounted for by introducing density jumps at compositional boundaries.

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