GEOP 529/Math 519 Student Projects, 2005
Review of Arnold NeumaierÕs paper: SOLVING ILL-CONDITIONED AND SINGULAR LINEAR SYSTEMS: A TUTORIAL ON REGULARIZATION
RAPHAEL CLANCY
Introduction
Reference: SIAM REV. Vol. 40, No. 3, pp. 636666, September 1998
1
Measure the adsorption-dependent refractive index of zeolite thin film grown on optical fiber by tunable laser
Juan Hui
A tunable laser is used to measure the refractive index change of a zeolite film caused by the environmental isopropanol concentration change. The laser is launched into a zeolite-based fiber sensor and the reflected energy from the sensor is a sinusoidal function of the wavelength of the tunable laser.
Least square fitting is used to fit the experimental data and results in a sinusoidal curve. The wavelength and energy of peak and valley are extracted from the fitted sinusoidal curve. A mathematical model based on the two beam optical interference is set up and specified to analysis the fitted data obtained above. At last, the fitted data is processed by the model. The refractive index and optical length are resulted to direct future sensor development.
Outlines:
3-D velocity structure modeling
Jana Stankova
The 3-D velocity structure modeling creates a non-unique and non-linear inverse problem, which is very sensitive to the quality of the input. It is based on an initial 1-D velocity structure model. Basically, we try to find the smoothest model (1-D) fitting the data that will result in least misfit, and apply it as an initial guess for the 3-D model. Then we keep iterating our results until there is no significant statistical difference in following models. However, since the 3-D modeling problem is non-unique, the calculations need more constraints. These include a prior knowledge of the area and station corrections.
Outline:
I. Intro
II. Input
1. station corrections
2. quality of data
3. quality of P and S arrival picks
III. 1-D model
1. inverse methods (calculations)
2. trade offs
IV. 3-D inverse modeling
1. goals
2. non-uniqueness, non-linearity issues
3. calculations
4. example (maybeÉ)
V. Model resolution
1. limitations
2. results
VI. Conclusion
David Baird
Cross-well electrical impedance tomography (EIT) will be used to map
types of objects in the Earth. Data will be synthesized, computing the forward
problem of EIT, and then the inverse problem will be computed and
checked with the original model The forward problem will be computed using
the finite element method (FEM) or boundary element method (BEM) numerical
methods. Various regularizations will be explored when computing
the inverse problem.
Generalized Inverses
Gabrielle Duncan
Generalized inverses have applications in a wide range of inverse problems.
There are several categories of generalized inverses, defined by the specific
properties they satisfy. One special type of generalized inverse is the Moore-
Penrose pseudoinverse. The pseudoinverse is distinct in that it satisfies all
four basic properties and, unlike other generalized inverses, it is unique. This
paper outlines the basic properties of generalized inverses, looks at several
examples, and discusses their application to inverse problems.
Outline
¥ What is a generalized inverse
Definition
Properties (1-4)
Examples (e.g. 1-inverse, 1,2-inverse, etc.)
¥ Moore-Penrose pseudoinverse
Brief history
Definition
Show: satisfies properties 1-4
Significance: unique
¥ Applications of generalized inverses
Source Parameters of the August 10, 2005 Raton Basin Earthquake
From Empirical GreenÕs Functions
Tara Mayeau
Empirical GreenÕs Functions (EGF) are used in earthquake seismology to determine source parameters of earthquakes, such as the fault rupture processes and direction of rupture propagation. The method of EGF considers a seismogram as a convolution of a source function, a function of the geologic structure along the wave path, and an instrument response function. By solving for the source function for seismographs at different azimuths, it is possible to get a record of complex rupture patterns. Here this technique is applied to a magnitude 4.5 earthquake which occurred near Raton, New Mexico on August 10, 2005. However, this procedure is complicated in this case by a lack of good station coverage near the epicenter. The records from three seismic stations at different azimuths are analyzed to look for a possible second event, complex rupture patterns, and to determine the direction of rupture propagation.
Outline
NMR Spectral Fitting
Don Clewett
In fitting Nuclear Magnetic Resonance (NMR)spectra, the question often
arises with solid materials as to how many lines (Gaussian or
Lorentzian)should be fit to a particular spectrum, and as the lines often
overlap, is there an optimum number of lines to fit to a spectrum. I will
examine the Akaike Information Criterion (AIC), Schwartz Bayesian
Information Criterion (BIC) and (possibly) the bias adjusted AIC (AICC)
to see if they help to optimize the numbers of lines to fit, as well as
the other parameters of the model for the NMR signal.
Outline:
I. Intro
A. Brief intro to NMR and spectra.
B. Why Gaussians or Lorentzians.
C. The fitting problem and overlap.
II. AIC
A. What is the AIC
B. How does it work
C. Bias and limitations
III. BIC
A. What is the BIC
B. How to use it
C. Bias and limitations
IV. Bias Adjusted AIC
A. What is it.
B. How does it work.
C. Why is it better than the AIC.
V. Actual Example with Spectra from 131Xe study of Carbon Nanotubes
A. Compare the possible models
B. Apply the criteria
C. Discuss results.
VI. Conclusion