A continuation approach to regularization of ill-posed problems with application to crosswell-traveltime tomography

Geophysics ◽  
2008 ◽  
Vol 73 (5) ◽  
pp. VE337-VE351 ◽  
Author(s):  
Kenneth P. Bube ◽  
Robert T. Langan
Keyword(s):  
A Priori ◽  
Real Data ◽  
Inversion Algorithm ◽  
Surface Reflection ◽  
Traveltime Tomography ◽  
Ill Posed ◽  
Priori Information ◽  
The Many ◽  
Derivatives Of

In most geometries in which seismic-traveltime tomography is applied (e.g., crosswell, surface-reflection, and VSP), determination of the slowness field using only traveltimes is not a well-conditioned problem. Nonuniqueness is common. Even when the slowness field is uniquely determined, small changes in measured traveltimes can cause large errors in the computed slowness field. A priori information often is available — well logs, initial rough estimates of slowness from structural geology, etc. — and can be incorporated into a traveltime-inversion algorithm by using penalty terms. To further regularize the problem, smoothing constraints also can be incorporated using penalty terms by penalizing derivatives of the slowness field. What weights to use on the penalty terms is a major decision, particularly the smoothing-penalty weights. We use a continuation approach in selecting the smoothing-penalty weights. Instead of using fixed smoothing-penalty weights, we decrease them step by step, using the slowness model computed with the previous, larger weights as the initial slowness model for the next step with the new, smaller weights. This continuation approach can solve synthetic problems more accurately than does one that uses fixed smoothing-penalty weights, and it appears to yield more features of interest in real-data applications of traveltime tomography. We have formulated guidelines for making the many choices needed to implement this continuation strategy effectively and have developed specific choices for crosswell-traveltime tomography.

Resolution in crosswell traveltime tomography: The dependence on illumination

Geophysics ◽  
2016 ◽  
Vol 81 (1) ◽  
pp. W1-W12 ◽  
Author(s):  
Renato R. S. Dantas ◽  
Walter E. Medeiros
Keyword(s):  
A Priori ◽  
Synthetic Data ◽  
Geometric Approach ◽  
Real Data ◽  
Angular Aperture ◽  
Traveltime Tomography ◽  
Sparsity Constraints ◽  
Ill Posed ◽  
Priori Information ◽  
The One

The key aspect limiting resolution in crosswell traveltime tomography is illumination, a well-known result but not well-exemplified. We have revisited resolution in the 2D case using a simple geometric approach based on the angular aperture distribution and the Radon transform properties. We have analytically found that if an isolated interface had dips contained in the angular aperture limits, it could be reconstructed using just one particular projection. By inversion of synthetic data, we found that a slowness field could be approximately reconstructed from a set of projections if the interfaces delimiting the slowness field had dips contained in the available angular apertures. On the one hand, isolated artifacts might be present when the dip is near the illumination limit. On the other hand, in the inverse sense, if an interface is interpretable from a tomogram, there is no guarantee that it corresponds to a true interface. Similarly, if a body is present in the interwell region, it is diffusely imaged, but its interfaces, particularly vertical edges, cannot be resolved and additional artifacts might be present. Again, in the inverse sense, there is no guarantee that an isolated anomaly corresponds to a true anomalous body, because this anomaly could be an artifact. These results are typical of ill-posed inverse problems: an absence of a guarantee of correspondence to the true distribution. The limitations due to illumination may not be solved by the use of constraints. Crosswell tomograms derived with the use of sparsity constraints, using the discrete cosine transform and Daubechies bases, essentially reproduce the same features seen in tomograms obtained with the smoothness constraint. Interpretation must be done taking into consideration a priori information and the particular limitations due to illumination, as we have determined with a real data case.

An optimal regularization method for convolution equations on the sourcewise represented set

2015 ◽  
Vol 23 (5) ◽  
Author(s):  
Ye Zhang ◽  
Dmitry V. Lukyanenko ◽  
Anatoly G. Yagola
Keyword(s):  
Integral Equation ◽  
Regularization Method ◽  
A Priori ◽  
Noisy Data ◽  
Optimal Recovery ◽  
Multidimensional Case ◽  
A Priori Information ◽  
Ill Posed ◽  
Priori Information ◽  
Convolution Equations

AbstractIn this article, we consider an inverse problem for the integral equation of the convolution type in a multidimensional case. This problem is severely ill-posed. To deal with this problem, using a priori information (sourcewise representation) based on optimal recovery theory we propose a new method. The regularization and optimization properties of this method are proved. An optimal minimal a priori error of the problem is found. Moreover, a so-called optimal regularized approximate solution and its corresponding error estimation are considered. Efficiency and applicability of this method are demonstrated in a numerical example of the image deblurring problem with noisy data.

scholarly journals A procedure for meaningful unsupervised clustering and its application for solvent classification

2014 ◽  
Vol 12 (5) ◽  
pp. 594-603 ◽  
Author(s):  
Yaroslava Pushkarova ◽  
Yuriy Kholin
Keyword(s):  
Neural Networks ◽  
A Priori ◽  
Initial Distribution ◽  
C60 Fullerene ◽  
Classification Problems ◽  
Probabilistic Networks ◽  
Reliable Classification ◽  
Priori Information ◽  
Number Of Classes

AbstractArtificial neural networks have proven to be a powerful tool for solving classification problems. Some difficulties still need to be overcome for their successful application to chemical data. The use of supervised neural networks implies the initial distribution of patterns between the pre-determined classes, while attribution of objects to the classes may be uncertain. Unsupervised neural networks are free from this problem, but do not always reveal the real structure of data. Classification algorithms which do not require a priori information about the distribution of patterns between the pre-determined classes and provide meaningful results are of special interest. This paper presents an approach based on the combination of Kohonen and probabilistic networks which enables the determination of the number of classes and the reliable classification of objects. This is illustrated for a set of 76 solvents based on nine characteristics. The resulting classification is chemically interpretable. The approach proved to be also applicable in a different field, namely in examining the solubility of C60 fullerene. The solvents belonging to the same group demonstrate similar abilities to dissolve C60. This makes it possible to estimate the solubility of fullerenes in solvents for which there are no experimental data

Using a Priori Information in Energy-Dispersive X-Ray Fluorescence Analysis of Complex Samples

1991 ◽  
Vol 35 (B) ◽  
pp. 1205-1209
Author(s):  
I. A. Kondurov ◽  
P. A. Sushkov ◽  
T. M. Tjukavina ◽  
G. I. Shulyak
Keyword(s):  
High Temperature Superconductors ◽  
A Priori ◽  
Fluorescence Analysis ◽  
A Priori Information ◽  
Complex Samples ◽  
X Ray ◽  
Edxrf Analysis ◽  
The Matrix ◽  
Priori Information

In multielement EDXRF analysis of very complex unknowns, some problems in data evaluation may be simplified if one can take into account a priori information on the properties of the incident and detected radiations, and also available data on the matrix of the sample. The number of variables can be drastically shortened in the LSM procedures in this case. One of the best examples of complex unknowns is the determination of the rare earth element content of ores, and most recently in samples of high temperature superconductors (HiTc).

Estimation of density, magnetization, and depth to source: A nonlinear and noniterative 3-D potential‐field method

Geophysics ◽  
1997 ◽  
Vol 62 (3) ◽  
pp. 814-830 ◽  
Author(s):  
Maurizio Fedi
Keyword(s):  
A Priori ◽  
Real Data ◽  
Field Method ◽  
Magnetic Data ◽  
System Of Equations ◽  
Source Depth ◽  
A Priori Information ◽  
Magnetization Direction ◽  
Priori Information ◽  
Magnetic Case

The depth to the top, or bottom, and the density of a 3-D homogeneous source can be estimated from its gravity or magnetic anomalies by using a priori information on the maximum and minimum source depths. For the magnetic case, the magnetization direction is assumed to be constant and known. The source is assumed to be within a layer of known depth to the top h and thickness t. A depth model, satisfying both the data and the a priori information is found, together with its associated density/magnetization contrast. The methodology first derives, from the measured data, a set of apparent densities [Formula: see text] (or magnetizations), which do not depend on the layer parameters h and t, but only on source thickness. A nonlinear system of equations based on [Formula: see text], with source thicknesses as unknowns, is constructed. To simplify the solution, a more practical system of equations is formed. Each equation depends on only one value of thickness. Solving for the thicknesses, taking into account the above a priori information, the source depth to the top (or to the bottom) is determined uniquely. Finally, the depth solutions allow a unit‐density gravity model to be computed, which is compared to the observed gravity to determine the density contrast. A similar procedure can be used for magnetic data. Tests on synthetic anomalies and on real data demonstrate the good performance of this method.

scholarly journals Unconstrained, robust Kalman filtering for GNSS troposphere tomography

2013 ◽  
Vol 6 (5) ◽  
pp. 9133-9162 ◽  
Author(s):  
W. Rohm ◽  
K. Zhang ◽  
J. Bosy
Keyword(s):  
Water Vapour ◽  
Kalman Filtering ◽  
Weather Prediction ◽  
A Priori ◽  
Real Data ◽  
Reference Station ◽  
A Priori Information ◽  
Dependent Observations ◽  
Integrated Water Vapour ◽  
Priori Information

Abstract. The mesoscale variability of water vapour (WV) in the troposphere is a highly complex phenomenon and modeling and monitoring the WV distribution is a very important but challenging task. Any observation technique that can reliably provide WV distribution is essential for both monitoring and predicting weather. GNSS tomography technique is a powerful tool that builds upon the critical ground-based GNSS infrastructure – Continuous Operating Reference Station (CORS) networks and can be used to sense the amount of WV. Previous research suggests that 3-D WV field from GNSS tomography has an uncertainty of 1 hPa. However all the models used in GNSS tomography heavily rely on a priori information and constraints from non-GNSS measurements. In this study, 3-D GNSS tomography models are investigated based on an unconstrained approach with limited a priori information. A case study is designed and the results show that unconstrained solutions are feasible by using a robust Kalman filtering technique and effective removal of linearly dependent observations and parameters. Discrepancies between reference wet refractivity data derived from the Australian Numerical Weather Prediction (NWP) model (i.e. ACCESS) and the GNSS tomography model using both simulated and real data are 4.2 ppm (mm km−1) and 6.5 ppm (mm km−1), respectively, which are essentially in the same order of accuracy. Therefore the accuracy of the integrated values should not be worse than 0.06 m in terms of zenith wet delay and the integrated water vapour is a fifth of this value which is roughly 10 mm.

scholarly journals A High-Throughput Analysis Method to Detect Regions of Interest and Quantify Zebrafish Embryo Images

2010 ◽  
Vol 15 (9) ◽  
pp. 1152-1159 ◽  
Author(s):  
Xiaoyan Xu ◽  
Xiaoyin Xu ◽  
Xin Huang ◽  
Weiming Xia ◽  
Shunren Xia
Keyword(s):  
High Throughput ◽  
Neural Development ◽  
Zebrafish Embryo ◽  
A Priori ◽  
Region Of Interest ◽  
Real Data ◽  
Zebrafish Embryos ◽  
High Throughput Analysis ◽  
Single Experiment ◽  
Priori Information

Zebrafish is widely used to understand neural development and model various neurodegenerative diseases. Zebrafish embryos are optically transparent, have a short development period, and can be kept alive in microplates for days, making them amenable to high-throughput microscopic imaging. As a result of high-throughput experiments, a large number of images can be generated in a single experiment, posing a challenge to researchers to analyze them efficiently and quantitatively. In this work, we develop an image processing focused on detecting and quantifying pigments in zebrafish embryos. The algorithm automatically detects a region of interest (ROI) enclosing an area around the pigments and then segment the pigments for quantification. In this process, the algorithm identifies the head and torso at first, and then finds the boundaries corresponding to the back and abdomen by taking advantage of a priori information about the anatomy of zebrafish embryos. The method is robust in terms that it can detect and quantify pigments even when the embryos have different orientations and curvatures. We used real data to demonstrate the performance of the method to extract phenotypic information from zebrafish embryo images and compared its results with manual analysis for verification.

scholarly journals Retrieval of Canopy component temperatures through Bayesian inversion of directional thermal measurements

2009 ◽  
Vol 6 (2) ◽  
pp. 3007-3040 ◽  
Author(s):  
J. Timmermans ◽  
W. Verhoef ◽  
C. van der Tol ◽  
Z. Su
Keyword(s):  
Sensitivity Analysis ◽  
Surface Temperature ◽  
A Priori ◽  
Inversion Method ◽  
Success Rates ◽  
Brightness Temperatures ◽  
Ill Posed ◽  
Priori Information ◽  
The Individual ◽  
High Level

Abstract. In remote sensing evapotranspiration is estimated using a single surface temperature. This surface temperature is an aggregate over multiple canopy components. The temperature of the individual components can differ significantly, introducing errors in the evapotranspiration estimations. The temperature aggregate has a high level of directionality. An inversion method is presented in this paper to retrieve four canopy component temperatures from directional brightness temperatures. The Bayesian method uses both a priori information and sensor characteristics to solve the ill-posed inversion problem. The method is tested using two case studies: 1) a sensitivity analysis, using a large forward simulated dataset, and 2) in a reality study, using two datasets of two field campaigns. The results of the sensitivity analysis show that the Bayesian approach is able to retrieve the four component temperatures from directional brightness temperatures with good success rates using multi-directional sensors (ℜspectra≈0.3, ℜgonio≈0.3, and ℜAATSR≈0.5), and no improvement using mono-angular sensors (ℜ≈1). The results of the experimental study show that the approach gives good results for high LAI values (RMSEgrass=0.50 K, RMSEwheat=0.29 K, RMSEsugar beet=0.75 K, RMSEbarley=0.67 K), but for low LAI values the measurement setup provides extra disturbances in the directional brightness temperatures, RMSEyoung maize=2.85 K, RMSEmature maize=2.85 K. As these disturbances, were only present for two crops and can be eliminated using masked thermal images the method is considered successful.

scholarly journals Retrieval of canopy component temperatures through Bayesian inversion of directional thermal measurements

2009 ◽  
Vol 13 (7) ◽  
pp. 1249-1260 ◽  
Author(s):  
J. Timmermans ◽  
W. Verhoef ◽  
C. van der Tol ◽  
Z. Su
Keyword(s):  
Sensitivity Analysis ◽  
A Priori ◽  
Inversion Method ◽  
Success Rates ◽  
Brightness Temperatures ◽  
Single Temperature ◽  
Ill Posed ◽  
Priori Information ◽  
The Individual ◽  
High Level

Abstract. Evapotranspiration is usually estimated in remote sensing from single temperature value representing both soil and vegetation. This surface temperature is an aggregate over multiple canopy components. The temperature of the individual components can differ significantly, introducing errors in the evapotranspiration estimations. The temperature aggregate has a high level of directionality. An inversion method is presented in this paper to retrieve four canopy component temperatures from directional brightness temperatures. The Bayesian method uses both a priori information and sensor characteristics to solve the ill-posed inversion problem. The method is tested using two case studies: 1) a sensitivity analysis, using a large forward simulated dataset, and 2) in a reality study, using two datasets of two field campaigns. The results of the sensitivity analysis show that the Bayesian approach is able to retrieve the four component temperatures from directional brightness temperatures with good success rates using multi-directional sensors (Srspectra≈0.3, Srgonio≈0.3, and SrAATSR≈0.5), and no improvement using mono-angular sensors (Sr≈1). The results of the experimental study show that the approach gives good results for high LAI values (RMSEgrass=0.50 K, RMSEwheat=0.29 K, RMSEsugar beet=0.75 K, RMSEbarley=0.67 K); but for low LAI values the results were unsatisfactory (RMSEyoung maize=2.85 K). This discrepancy was found to originate from the presence of the metallic construction of the setup. As these disturbances, were only present for two crops and were not present in the sensitivity analysis, which had a low LAI, it is concluded that using masked thermal images will eliminate this discrepancy.

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