Enhancing the resolving power of the least squares inversion with active constraint balancing

1998 ◽  
Author(s):  
Myeong‐Jong Yi ◽  
Jung‐Ho Kim
Keyword(s):  
Least Squares ◽  
Resolving Power ◽  
Active Constraint

Enhancing the resolving power of least‐squares inversion with active constraint balancing

Geophysics ◽  
2003 ◽  
Vol 68 (3) ◽  
pp. 931-941 ◽  
Author(s):  
Myeong‐Jong Yi ◽  
Jung‐Ho Kim ◽  
Seung‐Hwan Chung
Keyword(s):  
Least Squares ◽  
Function Analysis ◽  
Resolution Enhancement ◽  
Resolving Power ◽  
Stability And Convergence ◽  
Lagrangian Multiplier ◽  
Active Constraint ◽  
Data Set ◽  
Resistivity Tomography ◽  
The Stability

Most geophysical inverse problems are solved using least‐squares inversion schemes with damping or smoothness constraints to improve stability and convergence rate. Since the Lagrangian multiplier controls resolution and stability of the inverse problem, we always want to use the optimum multiplier, which is not easy to get and is usually obtained by experience or a time‐consuming optimization process. We present a new regularization approach, in which the Lagrangian multiplier is set as a spatial variable at each parameterized block and automatically determined via the parameter resolution matrix and spread function analysis. For highly resolvable parameters, a small value of the Lagrangian multiplier is assigned, and vice versa. This approach, named “active constraint balancing” (ACB), tries to balance the constraints of the least‐squares inversion according to sensitivity for a given problem so that it enhances the resolution as well as the stability of the inversion process. We demonstrate the performance of the ACB by applying it to a two‐dimensional resistivity tomography problem, which results in a remarkable enhancement of the spatial resolution. Enhancement of the resolution is also verified in the application of resistivity tomography to a field data set acquired at a tunnel construction site.

scholarly journals Statistical precision of the intensities retrieved from constrained fitting of overlapping peaks in high-resolution mass spectra

2015 ◽  
Vol 8 (6) ◽  
pp. 2333-2345 ◽  
Author(s):  
M. J. Cubison ◽  
J. L. Jimenez
Keyword(s):  
High Resolution ◽  
Least Squares ◽  
Resolving Power ◽  
Least Squares Fitting ◽  
Peak Separation ◽  
Peak Intensity ◽  
High Resolution Mass ◽  
Statistical Precision ◽  
Overlapping Peaks ◽  
Resolution Mass

Abstract. Least-squares fitting of overlapping peaks is often needed to separately quantify ions in high-resolution mass spectrometer data. A statistical simulation approach is used to assess the statistical precision of the retrieved peak intensities. The sensitivity of the fitted peak intensities to statistical noise due to ion counting is probed for synthetic data systems consisting of two overlapping ion peaks whose positions are pre-defined and fixed in the fitting procedure. The fitted intensities are sensitive to imperfections in the m/Q calibration. These propagate as a limiting precision in the fitted intensities that may greatly exceed the precision arising from counting statistics. The precision on the fitted peak intensity falls into one of three regimes. In the "counting-limited regime" (regime I), above a peak separation χ ~ 2 to 3 half-widths at half-maximum (HWHM), the intensity precision is similar to that due to counting error for an isolated ion. For smaller χ and higher ion counts (~ 1000 and higher), the intensity precision rapidly degrades as the peak separation is reduced ("calibration-limited regime", regime II). Alternatively for χ < 1.6 but lower ion counts (e.g. 10–100) the intensity precision is dominated by the additional ion count noise from the overlapping ion and is not affected by the imprecision in the m/Q calibration ("overlapping-limited regime", regime III). The transition between the counting and m/Q calibration-limited regimes is shown to be weakly dependent on resolving power and data spacing and can thus be approximated by a simple parameterisation based only on peak intensity ratios and separation. A simple equation can be used to find potentially problematic ion pairs when evaluating results from fitted spectra containing many ions. Longer integration times can improve the precision in regimes I and III, but a given ion pair can only be moved out of regime II through increased spectrometer resolving power. Studies presenting data obtained from least-squares fitting procedures applied to mass spectral peaks should explicitly consider these limits on statistical precision.

A Nonlinear Least Squares Method of Determining Line Intensities and Half-Widths

1977 ◽  
Vol 31 (3) ◽  
pp. 213-220 ◽  
Author(s):  
Y. S. Chang ◽  
J. H. Shaw
Keyword(s):  
Least Squares ◽  
Least Squares Method ◽  
Nonlinear Least Squares ◽  
Resolving Power ◽  
Line Intensities ◽  
Nonlinear Least Squares Method

A nonlinear least squares method of retrieving line intensities and half-widths from spectra degraded by a finite instrumental resolving power is discussed. Examples of the applications of this technique to obtain the parameters of a single isolated line and of an overlapping pair of lines are presented.

scholarly journals On the Aliasing and Resolving Power of Sea Level Low-Pass Filtered onto a Regular Grid from Along-Track Altimeter Data of Uncoordinated Satellites: The Smoothing Strategy

2008 ◽  
Vol 25 (4) ◽  
pp. 617-624
Author(s):  
Chang-Kou Tai
Keyword(s):  
Least Squares ◽  
Resolving Power ◽  
Altimeter Data ◽  
Spectral Space ◽  
Combining Data ◽  
Low Pass ◽  
Spectral Spaces ◽  
Band Limited ◽  
Along Track ◽  
Least Squares Approach

Abstract It is shown that smoothing (low-pass filtering) along-track altimeter data of uncoordinated satellites onto a regular space–time grid helps reduce the overall energy level of the aliasing from the aliasing levels of the individual satellites. The rough rule of thumb is that combining N satellites reduces the energy of the overall aliasing to 1/N of the average aliasing level of the N satellites. Assuming the aliasing levels of these satellites are roughly of the same order of magnitude (i.e., assuming that no special signal spectral content significantly favors one satellite over others at certain locations), combining data from uncoordinated satellites is clearly the right strategy. Moreover, contrary to the case of coordinated satellites, this reduction of aliasing is not achieved by the enhancement of the overall resolving power. In fact (by the strict definition of the resolving power as the largest bandwidths within which a band-limited signal remains free of aliasing), the resolving power is reduced to its smallest possible extent. If one characterizes the resolving power of each satellite as a spectral space within which all band-limited signals are resolved by the satellite, then the combined resolving power of the N satellite is characterized by the spectral space that is the intersection of all N spectral spaces (i.e., the spectral space that is common to all the resolved spectral spaces of the N satellites, hence the smallest). It is also shown that the least squares approach is superior to the smoothing approach in reducing the aliasing and upholding the resolving power of the raw data. To remedy one of the shortcomings of the smoothing approach, the author recommends a multismoother smoothing strategy that tailors the smoother to the sampling characteristics of each satellite. Last, a strategy based on the least squares approach is also described for combining data from uncoordinated satellites.

scholarly journals Statistical precision of the intensities retrieved from constrained fitting of overlapping peaks in high-resolution mass spectra

2014 ◽  
Vol 7 (12) ◽  
pp. 12617-12647 ◽  
Author(s):  
M. J. Cubison ◽  
D. Sueper ◽  
J. L. Jimenez
Keyword(s):  
High Resolution ◽  
Least Squares ◽  
Resolving Power ◽  
Least Squares Fitting ◽  
Peak Separation ◽  
Peak Intensity ◽  
High Resolution Mass ◽  
Statistical Precision ◽  
Overlapping Peaks ◽  
Resolution Mass

Abstract. Least-squares fitting of overlapping peaks is often needed to separately quantify ions in high-resolution mass spectrometer data. A statistical simulation approach is used to assess the statistical precision of the retrieved peak intensities. The sensitivity of the fitted peak intensities to statistical noise due to ion counting is probed for synthetic data systems consisting of two overlapping ion peaks whose positions are pre-defined and fixed in the fitting procedure. The fitted intensities are sensitive to imperfections in the m/Q calibration. These propagate as a limiting precision in the fitted intensities that may greatly exceed the precision arising from counting statistics. The precision on the fitted peak intensity falls into one of three regimes. In the "counting-limited regime" (regime I), above a peak separation χ ~ 2 to 3 half widths at half-maximum (HWHM), the intensity precision is similar to that due to counting error for an isolated ion. For smaller χ and higher ion counts (~1000 and higher), the intensity precision rapidly degrades as the peak separation is reduced ("calibration-limited regime", regime II). Alternatively for χ < 1.6 but lower ion counts (e.g. 10–100) the intensity precision is dominated by the additional ion count noise from the overlapping ion, and is not affected by the imprecision in the m/Q calibration ("overlapping-limited regime," regime III). The transition between the counting and m/Q calibration-limited regimes is shown to be weakly dependent on resolving power and data spacing and can thus be approximated by a simple parameterisation based only on peak intensity ratios and separation. An approximate graphical diagnostic can also be used to find potentially problematic ion pairs when evaluating results from fitted spectra containing many ions. Longer integration times can improve the precision in regimes I and III, but a given ion pair can only be moved out of regime II through increased spectrometer resolving power. Studies presenting data obtained from least-squares fitting procedures applied to mass spectral peaks should explicitly consider these limits on statistical precision.

Preparatory Procedures for Electron Microscopic Analysis at the Molecular Level of Resolution

1978 ◽  
Vol 36 (3) ◽  
pp. 516-520
Author(s):  
F.J. Sjostrand
Keyword(s):  
Electron Microscope ◽  
Molecular Level ◽  
Microscopic Analysis ◽  
Resolving Power ◽  
Cellular Structures ◽  
Electron Microscopic ◽  
Systematic Analysis ◽  
Technical Developments ◽  
Thin Sectioning ◽  
Obvious Reason

In the 1940's and 1950's electron microscopy conferences were attended with everybody interested in learning about the latest technical developments for one very obvious reason. There was the electron microscope with its outstanding performance but nobody could make very much use of it because we were lacking proper techniques to prepare biological specimens. The development of the thin sectioning technique with its perfectioning in 1952 changed the situation and systematic analysis of the structure of cells could now be pursued. Since then electron microscopists have in general become satisfied with the level of resolution at which cellular structures can be analyzed when applying this technique. There has been little interest in trying to push the limit of resolution closer to that determined by the resolving power of the electron microscope.

The High Resolution Scanning Transmission Electron Microscope

1974 ◽  
Vol 32 ◽  
pp. 316-317
Author(s):  
A. V. Crewe
Keyword(s):  
Electron Microscope ◽  
High Resolution ◽  
High Vacuum ◽  
Scanning Transmission Electron Microscope ◽  
Ultra High Vacuum ◽  
Resolving Power ◽  
Imaging Characteristics ◽  
Transmission Electron ◽  
Scanning Transmission ◽  
The Moment

The high resolution STEM is now a fact of life. I think that we have, in the last few years, demonstrated that this instrument is capable of the same resolving power as a CEM but is sufficiently different in its imaging characteristics to offer some real advantages.It seems possible to prove in a quite general way that only a field emission source can give adequate intensity for the highest resolution^ and at the moment this means operating at ultra high vacuum levels. Our experience, however, is that neither the source nor the vacuum are difficult to manage and indeed are simpler than many other systems and substantially trouble-free.

Transmission Electron Microscopy of Protein Macromolecules

1971 ◽  
Vol 29 ◽  
pp. 424-425
Author(s):  
Henry S. Slayter
Keyword(s):  
Biological Systems ◽  
Metal Vapor ◽  
Resolving Power ◽  
Electron Microscopic ◽  
Chemical Methods ◽  
Molecular Weights ◽  
The Past ◽  
Physical Chemical ◽  
Transmission Electron

Electron microscopic methods have been applied increasingly during the past fifteen years, to problems in structural molecular biology. Used in conjunction with physical chemical methods and/or Fourier methods of analysis, they constitute powerful tools for determining sizes, shapes and modes of aggregation of biopolymers with molecular weights greater than 50, 000. However, the application of the e.m. to the determination of very fine structure approaching the limit of instrumental resolving power in biological systems has not been productive, due to various difficulties such as the destructive effects of dehydration, damage to the specimen by the electron beam, and lack of adequate and specific contrast. One of the most satisfactory methods for contrasting individual macromolecules involves the deposition of heavy metal vapor upon the specimen. We have investigated this process, and present here what we believe to be the more important considerations for optimizing it. Results of the application of these methods to several biological systems including muscle proteins, fibrinogen, ribosomes and chromatin will be discussed.

The Broad Applications of the Scanning Electron Microscope

1969 ◽  
Vol 27 ◽  
pp. 20-21
Author(s):  
C. T. Nightingale ◽  
S. E. Summers ◽  
T. P. Turnbull
Keyword(s):  
Electron Microscope ◽  
Scanning Electron Microscope ◽  
Light Microscopy ◽  
Surface Topography ◽  
Great Depth ◽  
Resolving Power ◽  
Depth Of Field ◽  
Pictorial Representations ◽  
Scanning Electron

The ease of operation of the scanning electron microscope has insured its wide application in medicine and industry. The micrographs are pictorial representations of surface topography obtained directly from the specimen. The need to replicate is eliminated. The great depth of field and the high resolving power provide far more information than light microscopy.

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