Problem Set 3: Probabilistic inversion for earthquake epicenter
GEOS 627: Inverse Problems and Parameter Estimation, Carl Tape
Assigned: February 6, 2017 — Due: February 13, 2017
Last compiled: April 24, 2017
Overview and Instructions
• This is the first problem from the Problems chapter in Tarantola (2005). It is an example
of a problem that is computationally small enough such that the misfit function can be
evaluated almost anywhere in the space of model parameters. This allows for a purely
probabilistic solution to the problem.
Quoting Tarantola (2005, p. 38):
If the number of model parameters is very small (say, less than 10) and if the
computation of the numerical value of σM (m) for an arbitrary m is inexpensive
(i.e., not consuming too much computer time), we can define a grid over the model
space, compute σM (m) everywhere in the grid, and directly use these results to
discuss the information obtained on the model parameters. This is certainly the
most general way of solving the inverse problem. Problem 7.1 gives an illustration
of the method.
• Problem 2 involves sampling the probability distribution obtained in Problem 1. You will
use the rejection method of sampling, which is introduced in Section 2.3.2 of Tarantola
(2005) (“The Rejection Method”). This method was featured in lab_sampling.pdf. The
method is attributed to von Neumann (1951).
• For some homework problems, like this one, I provide a template script. Make a copy of
this script, then work with epicenter.m:
cp epicenter_template.m epicenter.m
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Problem 1 (4.0). Probabilistic solution, Part I
1. (0.7) Consider Tarantola’s Problem 7-1 (p. 253).
(a) (0.1) What are the unknown model parameters? How many?
(b) (0.1) What are the observations? How many?
(c) (0.1) What is the forward model (or “mathematical model”)? (What is the mathematical relationship between model parameters and the observations?)
(d) (0.3) Is the forward model linear or nonlinear? Why?
(e) (0.1) What parameters are assumed to be fixed (or constant)?
2. (3.5) Using Matlab, implement the misfit function of Tarantola’s Problem 7-1 (p. 253),
which can be written in the following ways:
F (m) =
=
=
2
6
1 X ti (m) − tobs
i
2
σ2
i=1
T 1 obs
obs
t(m)
−
t
t(m)
−
t
2σ 2
1 obs
obs
·
t(m)
−
t
t(m)
−
t
2σ 2
(1)
(2)
(3)
where σ, tobs
i , and the forward model ti (m) are provided in Tarantola (2005), and the model
vector represents the epicenter m = (xs , y s ). Here t(m) is a 6 × 1 vector of predicted arrival
times, and tobs is a 6 × 1 vector of observed arrival times.
(a) (2.8) Produce a colored 2D plot showing the F (m). This will involve evaluating F (m)
for a grid of m (see epicenter.m).
• For discretizing model parameter space, use the model parameter ranges x =
[0, 22] and y = [−2, 30].
• For all spatial plots in this problem set, use the following axes command,
axis equal, axis([0 22 -2 30])
(b) (0.5) Does the pattern of the misfit function seem reasonable, given the experiment?
Discuss.
Some tips:
• Be certain that you have entered the input parameters (e.g., station locations, travel times) correctly! (Your results will be wrong if you enter the input
incorrectly.)
• You may find it useful to define in-line functions (see lab_sampling.pdf). I have
provided epicenter.m as a starting point. This template includes code for generating
a set of points covering the 2D model space (you can change dx). Note that I have
already used variable names that you need to fill with numbers (e.g., tobs).
• The code in epicenter.m implies that you search over the grid of m using a single
for loop. This will store the vector of values fi = F (xi , yi ); in order to plot this, you
will need to apply a reshape command, as we did in lab_linefit.m. (Or you could
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use scatter) To use pcolor (or related commands), you need to provide same-size
matrices of X, Y, and F, as in pcolor(X,Y,F); shading flat. (See lab_linefit.m
for an example.)
• Plot your 6 stations on your plots. To do so, you need to type hold on after the figure
has been made (or after the first thing had been plotted).
• Since you will be varying the data vector, tobs , in Problem 3, you might want to
implement the misfit functions as F (m, tobs ).
Problem 2 (1.5). Probabilistic solution, Part II
Consider the posterior probability density (eq. 7.9)
σM (m) =
1 −F (m)
e
.
K
(4)
Assume that σM (m) = 0 outside our selected region of 2D model space.
1. (0.5) Using eq. 1.110,
Z
σM (m) dm,
P (A) =
(5)
A
derive an expression for K in terms of the variables in this problem.
Your expression for K should have the (unexpanded) term F (x, y) in it.
Hint: What is the probability that the epicenter is somewhere in our model domain?
2. (0.5)
(a) (0.3) Write your integral expression for K in discretized form (hint: use dA).
(b) (0.2) Compute σM (m) and plot a figure similar to Figure 7.1 of Tarantola (2005).
3. (0.3) List your values for K, max(σM ), and the grid increment dx. Include units.
4. (0.2) Try a different value for dx, and list the new calculations for K and max(σM ). Check
that your results do not change (much) when you change the grid increment dx.
dx, km
dA, km2
max[σM (m)]
K
0.1
0.2
1.0
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Problem 3 (2.5). Generating samples of the probabilistic solution
Consider the posterior probability density σM (m) from the previous problem. The goal of this
problem is to generate samples of σM (m), whereby a “sample” is a m = (xs , y s ) that has passed
the “rejection method” test.
1. (0.5) Make a sketch showing how you expect the samples to be distributed in the xy-plane.
(IN other words, plot a few dozen dots.)
If you’re confident that your algorithm in the next problem worked, then you can skip this
sketch.
2. (1.5) Generate samples of σM (m), and extract a subset of exactly 50 samples for the remaining analysis. Include a plot showing the samples in the xy-plane, and include the
relevant code. Use the same fixed axes scale as before.
Hints:
• For background on the rejection method, see lab_sampling.m and Tarantola (2005,
Section 2.3.2). (Note that the example in lab_sampling.m was for a function of one
variable, not two.)
• You will need to “try” some much larger number of samples (NTRY) in order to get at
least 50 samples that pass the test and are therefore representative of σM (m).
• Check that your NTRY test samples mi cover your full model space.
• Check that your evaluations of the test samples, σM (mi ), all range between 0 and 1.
• Plot the first 50 samples that pass the rejection test. If you do not get 50 samples
that pass the rejection test, try increasing NTRY.
3. (0.5) Perform the following statistical analysis on the ≥ 50 samples. (The trends look better
if you use more samples.)
(a) (0.2) Plot two histograms, one for xs and one for y s . (Consider using plot_histo.m)
Does either parameter appear to be Gaussian-distributed?
(b) (0.1) Compute the standard deviations of xs and y s . Do these quantities capture the
spread exhibited by the samples?
(c) (0.2) Compute the correlation coefficient (hint: corr) between xs and y s for the samples. Does this quantity capture the relationship between xs and y s ?
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Problem 4 (2.0). Consideration of data outliers and different norms
Here we revisit Problem 1. Use the grid discretization of dx = 0.2 km for this problem.
1. (0.5) Create an outlier measurement by setting the arrival time at the sixth station to be
1.0 s:
tobs
6 = 1.0 s
(6)
(a) (0.2) Solve for σM (m) with the new data vector, and include this plot.
(b) (0.1) How has the solution changed?
(c) (0.2) Why has it changed?
2. (1.0) Keep the data outlier, but compute the solution using a different norm for the misfit
function, the L1 norm:
F (m) =
6 X
ti (m) − tobs
i
i=1
(7)
σ
This has the same form as, for example, the first term of eq. 4.26 of Tarantola (2005), with
s = 1.
How does σM (m) compare with the solutions from Problems 2-4 and 4-1?
3. (0.5) For completeness, undo the data outlier (tobs
6 = 2.98 s) and solve for σM (m) using the
L1 misfit function. Briefly explain how the solution compares with the ones obtained in
Problems 2-4, 4-1, and 4-2. Summarize your results in Table 1.
Table 1: Summary of results for the four cases.
Problem
norm
outlier
2-4
L2
N
4-1
L2
Y
4-2
L1
Y
4-3
L1
N
dx, km
K
5
max[σM (m)]
Problem
Approximately how much time outside of class and lab time did you spend on this problem set?
Feel free to suggest improvements here.
References
Tarantola, A. (2005), Inverse Problem Theory and Methods for Model Parameter Estimation,
SIAM, Philadelphia, Penn., USA.
von Neumann, J. (1951), Various techniques used in connection with random digits, in Monte
Carlo Method, Applied Mathematics Series, vol. 12, pp. 36–38, U.S. Dept. Commerce Nat.
Bureau Standards, summary written by G. E. Forsythe.
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