# Lecture on Seismic Inversion

GEO 5210 – Seismology I L15-1
Lecture 15 – Seismic Inversion Part II
1. 3D Velocity Inversion – i.e., Travel-Time inversion or Global Tomography
Previously we talked about 1D velocity inversion. Now let’s talk about moving beyond 1D into 2- or 3-D.
Here, we seek v (x, z) or v (x, y, z), that is 2 or 3 dimensional structure. For example:
We may also choose to represent (“regularize”) our velocity at points instead of constant velocity cells:
Or…. we can use functional descriptions of model parameters. Common in global seismology are spherical harmonics, splines, Chebychev polynomials.
GEO 5210 – Seismology I L15-2
Important to remember
These velocity distributions are relative to some reference structure. For example:
It is good to first generally characterize our data set: travel time residuals δt (relative to some reference model, for example we can look at a histogram of them:
GEO 5210 – Seismology I L15-3
Are the travel time residuals not centered at δt = 0? Then the reference model may not be appropriate for these data, since typically we think of a reference model as fitting the average of the data.
Shape of the histogram may be due to:
– perturbations in velocity structure about the reference structure. e.g., if 3D, we can denote reference structure by v(z), and the perturbation field by δv(x, y, z). We define this as seismic heterogeneity
– errors in δt from mispicks of phases, noisy data, drunk seismologist, etc.
Let’s look at a 2-D example: a cross-section. We have some reference velocity structure and some distribution of sources and receivers:
We desire the perturbations in velocity in this cross-section, i.e., we want to know the deviations in velocity relative to the 1-D reference structure v(z) shown on the left –specifically, we want the perturbations in velocity that best predict the observed travel time behavior. We’ll define:
δv = velocity perturbation
vactual(x, z) = vref(z) + δv (x, z)
So, we want δv(x,z). Thus, we immediately see we have to make some decisions on how to set up our problem. The main issue here is the uneven distribution of ray path coverage. Each arrival time is compared to one predicted for our reference model. (Note that this requires assumption of an earthquake location and origin time.)
tobserved – tpredicted = δt
≡ travel time “anamoly” (or “perturbation”).
Let’s say we have 5 earthquakes
GEO 5210 – Seismology I L15-4
Obviously, for a non-zero travel time anomaly to exist, somewhere along the ray path the velocity structure must be different from our vref(z) structure. Then the approach is to use the observed δt’s to find that anomaly. Imagine this extremely simple case: one single constant velocity cell:
Then we have a fairly simple set up:
Travel time = tobs
Predicted time = tpred = vl
δt = tobserved – tpredicted
vvttlpredδδ+=+
Thus it is easy for us to get at the δv (just rearrange the above equation). That is in effect an inversion. Now let’s imagine a world full of such little cells, that is, a 2-D surface of them:
The choice of cell size would be based on our seismic ray path coverage density.
– excellent ray coverage? → relatively smaller cell size
– crappy ray coverage → relatively larger cell size
We already see issues that plague the tomographic approach:
– some regions may be well-sampled
– some regions may be poorly-sampled
Thus our answer (from the inversion) will be more meaningful in certain regions than in others. We could parameterize our model with variable “grid” (or cell) sizes, to reflect our ray-coverage. Doing so for some distribution of earthquakes and sources might yield:
GEO 5210 – Seismology I L15-5
Or, we can use functional forms if we wish, some spatial function, e.g., spherical harmonics. This approach has the advantage that error analysis and information about solution model heterogeneity scale length is much easier to obtain. However, there are known drawbacks to such regularizations too. The next image shows how splines are used to represent velocity versus depth (from Ritsema and van Heijst, 2000):
Let’s assume a Cartesian grid for our cross-section, and some ray path in this cross-section of cells:
GEO 5210 – Seismology I L15-6
Obviously, without redundant sampling of different cells, an inversion cannot determine uniquely which blocks are the sources of the any observed travel time anomaly. For example, you can imagine some distributed velocity perturbations completely along the path.
Let’s assume we have dense enough sampling of our medium to proceed. Then, one approach would be to tabulate the predicted lengths of the ray path in each cell. An algorithm might be as follows:
– Number or index the cells in the model (necessary book keeping)
– Assume on some reference velocity structure
– “Shoot” rays – down into the model
– Find the ray parameter of interest for each source-receiver combination
– Ray trace for each ray parameter and store the information on:
• Cell number of all cells crossed by each ray path
• The path length of all segments in each block (i.e., cell)
Let’s look at one cell, we might have for any given raypath:
As before, we have the time predicted (i.e., our reference model prediction) for the travel as
tref = refvl
But the perturbed time due to some velocity anomaly Δv would be:
tref + Δt = vvlrefΔ+
difference between difference between
tref and tobs (sec) vref and vactual (km/sec)
GEO 5210 – Seismology I L15-7
Δt = vvlrefΔ+─ refvl
= −Δ+1vvvvlrefrefref
And note that the vref + Δv = vactual, so that
Δt = tref −actactactrefvvvv
= tref 

−actactrefvvv
≡ δv
δv × 100 = % difference between
vref & vactual (unit-less)
Thus, for some ith block, we have a travel time perturbation, Δti:
Δti = ti δvi
where ti is the time in the ith block for the reference model at that depth, and δvi is the velocity anomaly in that block. Let’s do a simple version of the figure on the last page:
For the above path, use the individual cell path lengths and velocities in those cells to get an estimate of the total observed time:
obstvlvlvlvlvlvl=+++++5511111010998822
Or,
ttvvlvvlrefδδδ+=++++)(……)(555222.
GEO 5210 – Seismology I L15-8
Or, another way to formulate is in terms of residual times, the total travel time perturbation δt is:
558822……..vtvtvttδδδδ+++=
travel time of path velocity anomaly
accrued in block 8 in block 8
e.g. if δv8 =0.03 (meaning %31008=×−refrefvvv), and the path length in block 8 is 35 km, with a reference velocity of 7 km/sec, then the time of that leg of the ray path is 35 km divided by 7 km/sec = 5 sec. Thus the time anomaly is 5 sec times 0.03, or 0.15 sec.
So, our total travel time perturbation δt equation above can be more generally written (then more compactly) as:
Σ==+++=18118182211……..kkkvtvtvtvttδδδδδ
We can generalize for our ith observation: for our ith residual:
Σ==mijijivtt1δδ
Here, m = number of blocks in our model, and i = number of rays to be used. This can be written in matrix form:
=mimiimimivvvttttttttttttδδδδδδ……………………………………212122221121121
By now, you should know that we can invert this for the velocity perturbations. What do the results look like?
Images of global S-wave tomography models
GEO 5210 – Seismology I L15-9
Key to interpretation: Seismologists generally use the convention that we color seismic wave velocities that are lower than average Red, and wave velocities that are higher than average Blue. Originally this was because these changes in wave velocity were often interpreted as being due to temperature changes in the mantle (thus Red = slow = hot, and Blue = fast = cold). Unfortunately compositional variations also play a major role and this simple picture isn’t accurate. Note seismic wave velocities are shown as δv (i.e., percent difference from the 1D reference model).
Here are some cross-sections through the very first tomography models.
These models were a bit crude, but even by the late 1990’s topographers were starting to see what looked like fast (high velocity) anomalies associated with subducting slabs going all of the way down to the core-mantle boundary.
GEO 5210 – Seismology I L15-10
Here are some more images of cross-sections through subducted slabs.
Image courtesy of http://garnero.asu.edu. Several models are shown.
The next images show, map view, at different depths in the Earth.
GEO 5210 – Seismology I L15-11
An interesting observation is that the largest anomalies occur near the surface and near the core-mantle boundary. That is, the lithosphere and the CMB are highly heterogeneous!
At the surface we note a strong correlation between heat flow and tomography.
At the CMB we see two Large Low Shear Velocity Provinces (LLSVPs), red in the below image centered beneath the Pacific Ocean and Africa. These are surrounded by high velocity anomalies that are likely old subducted lithosphere (i.e., the CMB is a slab graveyard). The LLSVPs are especially interesting. They appear to have higher density than the surrounding mantle and exceptionally sharp sides. Both of those observations suggest these features are compositional anomalies.
GEO 5210 – Seismology I L15-12
Black lines indicate areas where studies have inferred sharp boundaries to the low velocities.
Interestingly, most hot spot volcanoes lie above the LLSVP’s, suggesting a link between them.
Locations of hot spot volcanoes are drawn with circles.
GEO 5210 – Seismology I L15-13
2. Finite Frequency Tomography
All of the previous discussion was based around using travel-times, and assuming that travel-time delays or advances occurred along the geometric ray. But, this isn’t strictly correct. Recall that we solve a problem of the form:
𝐀𝐀𝐀𝐀=𝐝𝐝
Where we define:
A = model/data kernel. This provides the relationship between model parameters and data.
In the previous case, the model/data kernel, just used a relationship based on the sensitivity of a seismic ray to a velocity perturbation. But, recall the case of a seismic wave encountering a low velocity anomaly:
Here the waves bend around the low-velocity anomaly. This is called wavefront healing. After a short distance after the waves have passed the anomaly, one observes almost no difference in the wavefront, almost as though the wave never even saw the anomaly.
https://home.chpc.utah.edu/~thorne/movies/Movie_Low_Velocity_Anomaly_1080p.wmv
The above animation shows a problem with ray theory. Low velocity anomalies are extremely difficult to detect, and ray theory is not entirely correct, in that the expected time delay of the ray passing right through the center of the low velocity anomaly is not as large as ray theory predicts.
To solve for this, seismologists have started using what are referred to as finite frequency sensitivity kernels. That is, instead of using the ray-based kernel we introduced above, we use a different kernel based on the full wave sensitivity to structures. As it turns out, seismic waves are most sensitive to low velocity anomalies in an area surrounding the geometric ray, as opposed to right along the ray itself! Weird? Indeed. When this was first introduced it sparked no small amount of controversy. The following figure shows an example of a finite frequency sensitivity kernel.
GEO 5210 – Seismology I L15-14
Red regions show the area of greatest sensitivity to low velocities. Note, how the most sensitive areas surround the ray path. The images on the right are looking right into the ray path. It should be obvious why these kernels are also referred to as banana-doughnut kernels.
In addition to using more physically realistic sensitivity kernels, recent efforts have also started using amplitude and phase information. i.e., more than just the travel-times recorded by seismic waves, but the full waveform. This gets termed full waveform tomography.
GEO 5210 – Seismology I L15-15
These advances, have at long last started to image whole mantle plumes as shown in the next figure from French & Romanowicz (2015).
The next figure shows the core-mantle boundary (CMB) from this model. Here they label things like ‘primary plumes’ which are resolved all of the way from the CMB to the surface. Note how the plumes are all associated with the deepest mantle low velocity provinces.
GEO 5210 – Seismology I L15-16

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