GEOP 529/Math 519 Student Projects, 2004
Qian Xia
L1 regression is an important method to solve inverse problems,
which is more outlier resistant than the least squares solution. L1
regression can be solved by IRLS method, but it is not suitable for very
large L1 regression problems. Interior point methods which are usually
used in LP are introduced to solve L1 regression and they are suitable for
solving this LP for very large problems.
Outlines:
1. Introduction of L1-regression problem
2. Introduction of Duality Theorem
3. How to reformulate the general L1-regression problem as the linear
programming problem
4. How to use interior point methods for solving this linear programming
for very large problem
5. Conclusion
GENERALIZED STRESS-STRAIN MODEL
John Moreno
A generalized stress-strain model is developed. Such model includes elastic, plastic, work hardening/softening, and failure considerations; and is to be proven with the aid of either experimental or synthetic data. The model construction starts from energy considerations, in order to develop the first version of the main equations. Then, some mathematical assumptions are made and the formal version of the main equations is obtained. Subsequently, elastic, plastic and failure principles are used in order to give the parameters of the model a more concise form. Finally, the model is tested with the aid of either experimental or synthetic data.
Calculation of electromagnetic sensitivities by the adjoint method
Nancy Natek
The goal of electromagnetic methods in geophysics is to determine the
distribution of electrical conductivity in the earth, and inverse theory
methods are used to extract this information from observational data. In
the adjoint method, sensitivities, or partial derivatives of an
electromagnetic field with respect to physical parameters, are obtained by
solving two boundary-value problems, one for a primary field due to a real
source, and a second for an auxiliary field due to a fictitious source.
Integration over the dot product of the fields in the region of interest
yields the desired sensitivity (McGillivray et al., 1994).
Introduction
- The primary objective of geophysical well logging for petroleum is to
identify potential reservoir rocks by determining their porosity and
permeability, and the nature and volume of fluids present. This can be
achieved by measuring an induced electromagnetic field produced by a
transmitter/receiver sonde.
Objective
- The objective of our modeling study is to investigate the 3D
electromagnetic induction problem for a single transmitter/receiver sonde
approaching a fault within a hydrocarbon reservoir.
Induction Sondes
- Description of tool to produce and measure electromagnetic fields in a
medium.
The Quarterspace Fault Model
- Statement of physical problem. Homogeneous medium with a single
conductivity vs. an inhomogeneous medium with a region of perturbed
conductivity.
The Adjoint Method
- Description of mathematical method for the solution of the sensitivity
in an inhomogeneous medium. Discussion of the data, model and solution
method.
Magnetic Field at the Receiver
- How to obtain the full magnetic field at the receiver of the sonde due
to a region with a perturbed conductivity.
Sensitivity Results
- Sensitivities calculated for various frequencies and sonde orientations.
Discussion of results.
Conclusions and Future Work
Inversion of Free Surface Reflection Data
Kent Anderson
I will be looking at the interference observed at the South Pole seismic station (QSPA) due to near vertically traveling seismic waves from earthquakes. The spectral scalloping observed can be related to the depth of the observations and the wave propagation characteristics (and thus the physical properties) of the snow and ice between the deep seismic instrument and the surface. I will attempt to use inversion techniques to back out these properties from the spectral data.
THE INVERSE SCATTERING PROBLEM
JORGE CISNEROS
Inverse scattering, or the inverse scattering problem, is the problem of
determining the characteristics of an object (its shape, internal
constitution, etc.) from measurement data of radiation or particles
scattered from the object. It is the inverse problem to the direct
scattering problem, which is determining the distribution of scattered
radiation/particle flux basing on the characteristics of the scatterer.
Since its early statement for radiolocation, the problem has found a vast
number of applications, such as echolocation, geophysical survey,
nondestructive testing, medical imaging, quantum field theory, to name
just a few.
Limitation of ForWARD algorithm in reconstructing interferometric images
Anandkumar Shetiya
The problem of deconvolution in the field of radio astronomy is a discrete ill-conditioned problem. Hence the task of designing deconvolution algorithms becomes much more harder. I tried to use an algorithm called ForWARD, to decompose an interferometric radio image into a set of scale sensitive basis functions and recontruct the true image of the radio source. ForWARD stands for Fourier-Wavelet Regularized Deconvolution. It is a hybrid deconvolution algorithm (Neelamani et al., IEEE Sig. Proc, 2003) that performs noise regularization via scalar shrinkage in Fourier as well as wavelet domains. I shall explain the Fourier-based Regularized Deconvolution (FoRD) part of the algorithm. This part was primarily responsible for its failure to solve the inverse problem of constructing the true image given the dirty image, the point spread function (psf) and random noise.
Key words - deconvolution, Tikhonov regularization, wavelets
A Synthetic Example of the HUT Retrieval Algorithm for Passive Microwave-Based Estimates of Snow Water Equivalent
Alex Rinehart
In their 2001 paper in Remote Sensing of Environment , J. Pullianen and Martti Hallikainen develop and implement an algorithm for the retrieval of snow water equivalent from 19GHz and 37GHz SSM/I radiometer observations. This algorithm, the Helsinki University of Technology (HUT) Model-based automatic inversion algorithm is examined and applied to observations of microwave temperature brightness in north central New Mexico to estimate snow water equivalent for specific dates in the 2000 snow season. The viability of the use of other analogous inverse methods is examined, and, where possible, employed. An analysis of the estimates follows.
I. Obtain SSM/I data for the 2000 snow season.
A. From NASA
B. From National Snow and Ice Center
II. Obtain and develop ancillary data
A. Land Use/Vegetative Fraction
1. USGS
2. STATSGO
B. Estimate of emissivity of snow and other EM quantities
1. Literature review
C. Snow Cover Fraction
1. Southwest RESAC
III. Find relation between relation between utilized parameters and SWE
1. Literature Review
IV. Develop MATLAB code to input and implement the HUT algorithm.
V. Examine viability of other least squares methods. If not too computationally expensive, implement.
V. Do basic error analysis, both statistical and physical.
Presentation Plan:
I. Introduction
A. What is SWE.
B. Why is SWE important
C. What is temperature brightness.
D. What is SSM/I
II. Physical model
A. Describe physical components of model
B. Go over major assumptions
III. Inverse Method
A. Derive cost function
B. Discuss algorithm used to solve
C. Discuss other methods that can be used.
IV. Analysis
A. Compare to AMSR-E data
B. Compare to SNOTEL sights
C. Discuss the effect of wet snow
D. Discuss the effect of no snow
E. Discuss the effect of vegetation parameter/snow parameter values
V. Conclusions
A. Is this effective?
B. What is this being used for?
C. How accurate is it?
D How efficient is it?
An Automatic, Adaptive Algorithm for Refining Phase Picks in Large Seismic Data Sets
Brent Henderson
I will be reviewing the inverse techniques used in the paper ÒAn Automatic, Adaptive Algorithm for Refining Phase Picks in Large Seismic Data SetsÓ by Rowe et al. To find the desired pick adjustment, the Polak-Ribiere conjugate gradient minimization is applied. This method provides a sparse G matrix, allowing this method to be used on large data sets. The conjugate gradient minimization is implemented using the minimum one norm residual solution (L1). The L1 solution provides a robust, outlier resistant solution for this problem. Outline 1) Reasons for finding an automated picking algorithm. 2) Preliminary data organization 3) Clustering 4) Estimation relative lag 5) Solving G*b=d a) Setting up G matrix b) Using the conjugate gradient method to solve for b. c) Implementation of the L1 solution for the conjugate gradient method d) Possible problems and solutions using the L1 solution e) Probability/rejecting outliers 6) Stack re-aligning and cross-correlation
Gravity and Potential Field Modeling
Rick Baars
Gravity modeling or potential field modeling has a couple of unique problems that need to be addressed.Ê I would take time to discuss the non-unique problems with potential fields modeling.Ê I will be attempting to invert Bouger gravity data that was collected around NMT campus.Ê The data would then be inverted using a method that would define the sedimentary basin contact with the basement rock.Ê A stable method for inversion has been developed that delineates sharp contact features and does not overly smooth the sediment basement contact.Ê I would like to discuss the methods used and present findings of the NMT survey.Ê
Outline
Intro to Gravity
ÊÊÊÊÊÊÊÊÊÊÊ Non-uniqueness
ÊÊÊÊÊÊÊÊÊÊÊ ÊÊÊÊÊÊÊÊÊÊÊ Density contrast
Gravity Data
ÊÊÊÊÊÊÊÊÊÊÊ NMT anomaly
Inversion Method
ÊÊÊÊÊÊÊÊÊÊÊ Discretization
ÊÊÊÊÊÊÊÊÊÊÊ Elementary prisms
ÊÊÊÊÊÊÊÊÊÊÊ Assumptions
ÊÊÊÊÊÊÊÊÊÊÊ Solving the inversion
Conclusion
Earthquake Source Time Function Inversion
Yingchun (Spring), Zhan
The determination of the source time function of an earthquake is a linear
inversion problem. Omnilinear inversion method is introduced in this
project, which is applied to determine the linear model parameters and
trace scaling factors to minimize the mismatch between observed and
synthetic seismograms. Larry J.Ruff, 1989. And also omnilinear inversion
can find the source time function and best-fit focal depth. By comparing
focal depths gotten by the omnilinear inversion and the standard linear
inversion, we can see the advantage of the omnilnear inversion. Also I
set data and use the omnilinear inversion to get the focal depths of
several earthquakes happened in Alaska-Aleutian.
The outline is:
1. Introduction
2. source time function deconvolution
3. omnilinear inversion and its application
4. the results from the omnilinear inversion and the standard inversion
5. the application of the omnilinear inversion in my research
Velocity to Pore Pressure Prediction
Osmar Rene Alcalde
An accurate knowledge of formation pore pressure is a key requirement for the safe and economic drilling of wells and understanding of reservoir performance. Several methods have been presented in the literature for this purpose. However, they mainly rely on empirical relations between velocity and pressure i.e., calibration of the model by means of log dataÑporosity, clay content, and sonic velocity logs (which sometimes are not available and/or expensive).
This proposal is intended to provide much of the tools necessary to address problems related to data integration in order to characterize petroleum reservoirs. Therefore, the objective is to use seismic velocity data to estimate reservoir pore pressure for a given set of rock properties, based on a theoretical rock-physics model without any attempt of using empirical relations. Consequently, a plausible reservoir pressure model should be obtained that allow us to characterize uncertainty in reservoir description (ultimately, this reservoir model may be used to evaluate the uncertainty in reservoir performance predictions under proposed operating conditions).
Rheological Characterization of Drilling Fluids Using a Temperature-Dependent Parameters Power Law Model.
Alejandro Wills
Accurate rheological characterization of fluids is extremely important in petroleum engineering, especially during drilling and production operations. Several models have been extensively used to describe the rheological behavior of fluids. One of them is the so-called power law, which can be written in terms of shearing stress, shear rate, consistency index, and behavior index. The two first variables are the measured data and the last two are the parameters to be found by inverse problem theory. The parameters as presented in this equation do not depend on pressure and temperature. When dealing with oil and gas wells spanning several thousands of feet underground, pressure and temperature become important and may affect the rheological properties of the fluids significantly. The purpose of this project is to include temperature dependency on the parameters and , and find a suitable method to invert the problem to characterize fluids in the best suitable form including temperature effects (pressure is not considered because of limitations in the available equipment). To accomplish this task several measurements of and at different temperatures will be made in the drilling laboratory at NMT (1 day), then I will try to define the best functions describing the parameters vs., temperature and include them in the original equation. Note that I can obtain parameters for every temperature measurement using simple linear regression, and then I can plot them vs. temperature and propose an equation that best describes their relationship (if it exist). The final step is to implement one of the techniques from the course to find the resulting parameters.