Matlab Primer, c/o Kermit Sigmon (pdf)
Example Past Class Projects (2005)
Example Past Class Projects (2004)
Parameter Estimation and Inverse Problems Textbook Homepage
Upper Mantle Attenuation Beneath the Rio Grande Rift and Colorado Plateau from the LA RISTRA Teleseismic Array, Southwestern United States
Jonathan MacCarthy, 2007
The lithosphere beneath the Rio Grande Rift and adjacent Colorado Plateau in the southwestern US has been subjected to regional thermal and chemical modifications of the Farallon subduction and later Miocene extension and magmatism. However, geochemical and seismological studies indicate that the Rift has undergone significant extension and mantle modification while the lithosphere beneath the Colorado Plateau has remained largely unchanged. Thermal and compositional differences beneath these two provinces provide important constraints on their divergent evolution since the Laramide. Attenuation of seismic energy is highly sensitive to temperature variations in the propagating medium. We measure relative attenuation from distant earthquakes that sample the lithosphere beneath the Rift and Plateau and invert for a 3D spatial model of attenuation]1, Q, to provide insight into variations of temperature versus composition beneath the provinces. Differential attenuation measurements are made by the method of Fourier spectral ratios, and the inversion uses raypaths and a preferred velocity model beneath the LA RISTRA teleseismic array.
GPS Source Inversion at Mount Erebus, Antarctica
Hunter Yarbrough, 2007
Mogi point and elliptical source modeling can be utilized for both forward and inverse modeling of volcanic sources. These techniques can provide estimates for many source parameters including the location and size of the source, which is of great interestat Mount Erebus volcano, Ross Island, Antarctica. Understanding the approximate location and size of a source can provide vital information for future equipment implementations. The goal of this project is to utilize modeling techniques for Mogi point and elliptical sources to best fit Global Positioning System (GPS) data obtained from Mount Erebus since 2001. For the initial investigation of this volcanic system, the time series created from the GPS data will not be used directly, but rather the velocity vectors will be obtained from these data for the inverse problem.This project also incorporates the use of the finite element modeling software, ABACUS. The primary goal of creating a finite element model is to generate half space models that can be utilized to provide information on the distribution of stresses and strains. ABACUS is being increasingly used in the geosciences, and has proved useful in modeling systems such as subduction zones.
Relaxation Time of a Polymer Chain
Taylor Dotson, 2007
The P2 function is examined in order to extract information about the relaxation time of a polymer chain in the hope that macroscopic behavior of that polymer can be inferred. P2 is the second Legendre polynomial autocorrelation function of u, where u is the vector drawn from one mer of the chain to another. An empirical model of the P2 function decay is given by: A*exp(t/T1), B*exp(t/T2)^beta, C*exp(t/T3), piecewise. Unfortunately, the change over points between the functions are uncertain. However, by the principle of time-temperature superposition one should be able to apply an appropriate horizontal scaling of a different temperature system, then be able to overlay its P2 function on that of another and therefore be able produce a more accurate estimate of the change over points. Data is obtained from LAMMPS classical molecular dynamic simulations and an optimized fit is obtained utilizing currently available programs.
X-ray Computed Tomography
James Hunter, 2007
X-ray computed tomography has been an important as a medical diagnostic technique for years and with cheap computer power is becoming increasingly common as an industrial non-destructive testing technique. After discussing the basic requirements of this type of algorithm, the first half of this talk will develop the parallel beam direct Fourier transform reconstruction from an initial line integral data set, using the Fourier slice theorem. A short pseudo-code describing the practical implementation of this algorithm will be presented. From here common point source geometries will be discussed with a particular emphasis on how these geometries modify the nature of the inverse problem and what issues these modifications can present. Data sufficiency and the common applications of the different geometries will also be discussed along with possible problems which can occur. To finalize several data sets and reconstructions will be shown to demonstrate the potential of x-ray computed tomography.
QR Factorization Least Squares Techniques
Josh Hanson, 2007
Porosity estimation from Acoustic Logging data
Soichiro Ota, 2007
The estimation of reservoir properties is considerably important factors for economically successful hydrocarbon exploration. Borehole geophysics is one of the major methodologies being applicable to geological and geophysical well evaluation. Several geophysical well logging tools are commonly widely used in hydrocarbon exploration and production phases to estimate essential reservoir properties.
In this class project, the P-wave and resistivity data acquired from geophysical logging tools are used to estimate important parameters for Archiefs equation such as cementation exponent (m), tortuosity factor (a) and water resistivity (Rw). The relationship between P-wave velocity-resistivity and these parameters can be described as non-linear behavior. Therefore, non-linear regression solution is applied.
Total Least Squares
Amanda North, 2007
I am going to introduce the topic of Total Least Squares which is a linear parameter estimation technique that is better for fitting data when there is an error in both the matrix and the vector in question. TLS is one of several fit techniques used when trying to estimate parameters of a linear multivariate problem. In my presentation I will compare the method of Total Least Squares to classical Least Squares, demonstrating how they differ. I will also give an example of TLS and the linear algebra used in this process.
Bounded Variable Least Squares
Aaron Wilson, 2007
The problem of minimizing the sum of squares of the elements of a variable subject to a set of linear inequality constraints on those variables is not uncommon. This type of problem is called a bounded variables least squares (BVLS) problem. Applications of a BVLS problem might include minimizing the residuals of an inverse problem solution given some constraints on the model parameters (these constraints may come from numerous sources, such as the physical meaning of the parameter). Efficient algorithms and implementations of those algorithms are necessary because the size of these problems can become very large when large data sets or numerous model parameters are used. There are several approaches to solving BVLS. MATLAB's lsqnonneg, while actually implementing only nonnegative constraints on the variables, will be used as a benchmark for these types of algorithms. It uses an active set method, attempting to determine which of the nonnegative constraints will be active (which of the inequalities will actually satisfy the equality constraint). This is a combinatorial approach to the problem and performs well even with poor conditioning. Another approach is the interior point method, a numerical approach that at its base uses Newton's method to traverse through the center of the feasible region. This can result in faster convergence, but often the problem becomes poorly conditioned when approaching the boundary of the feasible region and the optimal solution. Penalty functions and predictor-corrector methods are used to avoid this poor conditioning. I hope to compare MATALB's lsqnonneg to my implementation of an interior point method with a predictor corrector step, as well as implement optimizing features such as support for user defined matrix multiplication and observe the benefits (if any) on large problems with specific structures.
Homework 1: Due 2/2:
Exercises: 1.1, 1.2, A3, A5, A6, A7, A18, A19 from the textbook.
New Problem:
a) Write a Matlab subroutine, quadform(K), that diagonalizes the 2 by 2 positive definite matrix, K, and plots out its associated ellipsoidal bounding region (Equation A.84) using Delta = 1.
You may find the plotting code noted in Exercise 2.1(c), and using the "axis square" and "grid" commands when examining your plot to be helpful here.
b) Produce the appropriate plot for the matrix:
K= [ 2.5000 -0.5000 ; -0.5000 2.5000 ];
c) What are the exact semiaxis directions and lengths for your plotted ellipsoid in this case, and describe in words their relationship to the elements of the eigenvalue-eigenvector decomposition.
Homework 2: Due 2/16:
1. For the double-sided exponential distribution (B.8), find the expected
value and variance in terms of the parameters mu and sigma (please note erratum 9).
Exercises B.4, B.9, B.10, 2.1 (counts double), 2.5.
Note: Exercise 2.1 counts double relative to the other problems because it is much longer than the other exercises in this assignment.
Homework 3: Due 3/7:
Exercises 3.1, 4.2, 4.3 (counts double), 4.4. from the textbook
Homework 4: Due 3/23:
Exercises C.4, 5.2 (counts double) 6.2 a,b (part c for 10 points extra credit.), 7.2 from the textbook.
Homework 5: Due 4/16:
Exercises 8.1, 9.2 (counts double), and 9.3 from the textbook.
Last Updated March 5, 2007